Here's an answer from a non-particle physicist to complement what (former) professional particle physicist Anna V has written.
"Real particles" enter and leave Feynman diagrams. Therefore, in principle, they can be detected in an experiment - they are the "terminals" of a Feynman diagram: ports through which we can "see" the system within.
In contrast, the path of a virtual particle begins and ends within a Feynman diagram. It has no "free ends" dangling over the "boundaries" of the diagram and is therefore not directly measurable. We can't detect them in experiment.
None of this is likely new to you. You're still left wondering what reality we can ascribe to virtual particles, if we can't directly detect them. You can think of virtual particles more literally as Feynman liked to do, or you might try this approach: I personally like to think of them a little more abstractly as simply as mathematical terms in a perturbation series.
A good starting point to visualise this gist is the kinds of ideas explored in the following papers:
as well as the works of the late Hilary Booth of the Australian National University.
This is not standard QED and it is very specialised and contrived: think of it as an illustrative "Baby QED" for someone (like me) who hasn't mastered quantum field theory. We consider here the system of one electron, a proton (the latter thought of as a classical particle, simply setting up an inverse square electrostatic field in a Hydrogen atom and the "virtual photons" that are swapped between them. The electron in the classical potential is of course simply described by the first quantised Dirac equation. Now we add the electromagnetic field by adding Maxwell's equations and coupling the system as follows:
$$\gamma^\mu\left(i \partial_\mu - q A_\mu\right) \psi + V \psi - \psi = 0$$
$$\partial_\nu F^{\nu\,\mu} = q\,\bar{\psi} \gamma^\mu \psi$$
$$F_{\mu\,\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$
with the Lorenz Gauge
$$\partial_\mu A^\mu = 0$$.
The first equation is the Dirac equation, the second Maxwell's equations with a charge / current (4-current) distribution determined by the probability density of the Dirac electron. The third relates the Maxwell tensor (containing the $\vec{E}$ and $\vec{B}$ fields) to the four-potential, which couples back into the Dirac equation through the "gauge covariant derivative". So we have a rather elegant, but thorny to solve, coupled nonliear system.
In the papers, the equations lead to a fixed point problem $X=F(X)$ of a certain integro-differential operator $F$, which is contractive, so the solution is the limit of the sequence:
$$X_0,\, F(X_0),\, F^2(X_0),\,\cdots$$
and can thus be solved nonperturbatively, by the contraction mapping principle and it gives an infinite series of terms corresponding to virtual pairs too. It yields an exact solution which is an infinite series, what a mathematician would call the Peano-Baker series (see Baake and Schlaegel, "The Peano Baker Series" and it is what a theoretical particle physicist would call (I believe) the Dyson series.
Now the terms in this infinite series are $X_0$: Dirac's solution of the Hydrogen atom and the higher order terms are iterated integral operators: these iterations can be thought of as the perterbations wrought by one "virtual photon", the next term involves virtual photons and virtual pair production followed by virtual pair annihilation and so forth.
The "Virtual particles" in this viewpoint can be thought of simply as an evocative "mnemonic" to the structure of the mathematical terms in the infinite series.
Does the invariant helicity property contribute to the the concept of a photon and an "anti-photon" being the same entity?
Not really, and I think you're getting mixed up between helicity and chirality here. Take a look at this deep-water wave image by Kraaieniest. See how the red-dot test particles move in a helical-like fashion? They can't move "the other way" because the wave is what it is, and your own motion doesn't change that. Like Youstay was saying, a photon is a wave, and there is no such thing as a negative wave or an anti-wave or an anti-photon.
Are there any other properties unique to photons that I have not considered?
What you haven't considered is Dirac's belt, wherein "a Mobius strip is reminiscent of spin-1/2 particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state". When you make your photon go round and round a twisted Mobius path rather than move linearly, then the motion of the wave can have one of two chiralities. See the Mobius strip article on Wikipedia: "the Möbius strip is a chiral object with right- or left-handedness". Also see the Wikipedia spinor article, and there's the Mobius strip again:
Best Answer
The word photon is one of the most confusing and misused words in physics. Probably much more than other words in physics, it is being used with several different meanings and one can only try to find which one is meant based on the source and context of the message.
The photon that spectroscopy experimenter uses to explain how spectra are connected to the atoms and molecules is a different concept from the photon quantum optics experimenters talk about when explaining their experiments. Those are different from the photon that the high energy experimenters talk about and there are still other photons the high energy theorists talk about. There are probably even more variants (and countless personal modifications) in use.
The term was introduced by G. N. Lewis in 1926 for the concept of "atom of light":
As far as I know, this original meaning of the word photon is not used anymore, because all the modern variants allow for creation and destruction of photons.
The photon the experimenter in visible-UV spectroscopy usually talks about is an object that has definite frequency $\nu$ and definite energy $h\nu$; its size and position are unknown, perhaps undefined; yet it can be absorbed and emitted by a molecule.
The photon the experimenter in quantum optics (detection correlation studies) usually talks about is a purposely mysterious "quantum object" that is more complicated: it has no definite frequency, has somewhat defined position and size, but can span whole experimental apparatus and only looks like a localized particle when it gets detected in a light detector.
The photon the high energy experimenter talks about is a small particle that is not possible to see in photos of the particle tracks and their scattering events, but makes it easy to explain the curvature of tracks of matter particles with common point of origin within the framework of energy and momentum conservation (e. g. appearance of pair of oppositely charged particles, or the Compton scattering). This photon has usually definite momentum and energy (hence also definite frequency), and fairly definite position, since it participates in fairly localized scattering events.
Theorists use the word photon with several meanings as well. The common denominator is the mathematics used to describe electromagnetic field and its interaction with matter. Certain special quantum states of EM field - so-called Fock states - behave mathematically in a way that allows one to use the language of "photons as countable things with definite energy". More precisely, there are states of the EM field that can be specified by stating an infinite set of non-negative whole numbers. When one of these numbers change by one, this is described by a figure of speech as "creation of photon" or "destruction of photon". This way of describing state allows one to easily calculate the total energy of the system and its frequency distribution. However, this kind of photon cannot be localized except to the whole system.
In the general case, the state of the EM field is not of such a special kind, and the number of photons itself is not definite. This means the primary object of the mathematical theory of EM field is not a set of point particles with definite number of members, but a continuous EM field. Photons are merely a figure of speech useful when the field is of a special kind.
Theorists still talk about photons a lot though, partially because:
it is quite entrenched in the curriculum and textbooks for historical and inertia reasons;
experimenters use it to describe their experiments;
partially because it makes a good impression on people reading popular accounts of physics; it is hard to talk interestingly about $\psi$ function or the Fock space, but it is easy to talk about "particles of light";
partially because of how the Feynman diagram method is taught.
(In the Feynman diagram, a wavy line in spacetime is often introduced as representing a photon. But these diagrams are a calculational aid for perturbation theory for complicated field equations; the wavy line in the Feynman diagram does not necessarily represent actual point particle moving through spacetime. The diagram, together with the photon it refers to, is just a useful graphical representation of certain complicated integrals.)
Note on the necessity of the concept of photon
Many famous experiments once regarded as evidence for photons were later explained qualitatively or semi-quantitatively based solely based on the theory of waves (classical EM theory of light, sometimes with Schroedinger's equation added). These are for example the photoelectric effect, Compton scattering, black-body radiation and perhaps others.
There always was a minority group of physicists who avoided the concept of photon altogether for this kind of phenomena and preferred the idea that the possibilities of EM theory are not exhausted. Check out these papers for non-photon approaches to physics:
R. Kidd, J. Ardini, A. Anton, Evolution of the modern photon, Am. J. Phys. 57, 27 (1989) http://www.optica.machorro.net/Lecturas/ModernPhoton_AJP000027.pdf
C. V. Raman, A classical derivation of the Compton effect. Indian Journal of Physics, 3, 357-369. (1928) http://dspace.rri.res.in/jspui/bitstream/2289/2125/1/1928%20IJP%20V3%20p357-369.pdf
Trevor W. Marshall, Emilio Santos: The myth of the photon, Arxiv (1997) https://arxiv.org/abs/quant-ph/9711046v1
Timothy H. Boyer, Derivation of the Blackbody Radiation Spectrum without Quantum Assumptions, Phys. Rev. 182, 1374 (1969) https://dx.doi.org/10.1103/PhysRev.182.1374