For a spherical mirror, an object at the mirror's center has an image that is also at the center. Its magnification is $-1$. For a video showing this, see here.
If you stand slightly behind the center of a large spherical mirror and hold out your right hand, as if to shake, you will see a reflection of yourself, upside down, holding out its left hand. You can "touch" palm-to-palm with the reflection, or pass through it, but your hands won't be the in the "shaking" position. Instead, the thumbs point opposite directions.
Is it possible, using any combination of mirrors and lenses in the geometric optics limit, to create an image at the same location as the source object, with magnification $+1$, so that you could appear to be shaking your own hand? As you hold out your right hand, your image would need to appear right-side-up and also hold out its right hand.
Here is a faked picture of what I'm thinking of. To make it , I photographed my own hand, then used a computer to copy my hand, rotate it 180 degrees, line it up with the original, and "ghost" it out a bit.
There is no requirement that the image hand track yours if you move around – a static handshake is enough. I don't necessarily have to be able to view the image directly from where my head is, but it should be a real image so that if I make the air dusty, someone watching my hand from any angle would see the image.
Best Answer
Yes, it is possible.
Consider a mirror in the shape of a closed hemisphere - that is, half of a concave spherical mirror combined with a flat circular mirror passing through its center.
If you are inside this hemispherical mirror, you will see a real image of yourself rotated 180 degrees around the axis of the hemisphere. Here's one way to show this is true: The flat mirror creates a virtual image of you which is your reflection in the plane (duh), and the spherical mirror takes that as an object and creates a real image which is the inversion of that through the center of the sphere. The composition of the reflection and the inversion is a rotation.
Now, for practical reasons you don't want to be entirely enclosed by the mirror (it would be hard to get in and out, and you'd have to have your own light source...), so you could build just a part of the hemisphere. But the part you built would have to contain both some of the spherical surface and some of the flat surface for it to work.
If you have trouble understanding the orientation of everything, here is a concrete description: The flat mirror is a circle in the $z=0$ (horizontal) plane, $x^2+y^2\le1$. The spherical mirror is a hemisphere described by $z>0$, $x^2+y^2+z^2=1$. You're standing inside so your hand is at a point near the axis, for example, (0.01, 0, 0.2). What the mirrors do is this: For an object at a point $(x,y,z)$, the flat mirror creates an image at $(x,y,-z)$ (this image has opposite handedness because one coordinate is flipped). Then the spherical mirror takes that image at $(x,y,-z)$ and creates another image at $(-x,-y,z)$. This image has the same handedness as the original object. It is also "right side up" because the z coordinate is not flipped. The image of your right hand at (0.01, 0, 0.2) will appear as another right hand at (-0.01, 0, 0.2) which is in the appropriate orientation for you to shake hands.