Locate the line of reflection, and call it $L$. From the center of rotation, draw two lines such that the angle between the two lines is the angle of rotation, and such that $L$ bisects that angle.
When the rotation is first applied, it will carry one of the two lines you have drawn onto the other, and then when the reflection is applied, it will carry the line back to where it started. Whichever of those two lines it is, that is the line of reflection for the composition of the rotation with reflection.
You can reduce the second case to this case by noting that if $\sigma$ is a rotation and $\tau$ is a reflection, then you know that $\tau\sigma^{-1}$ is a reflection, so $\tau\sigma^{-1}=(\tau\sigma^{-1})^{-1}=\sigma \tau$. Since you've already understood where the line of reflection is for $\tau\sigma^{-1}$, that is exactly the line of rotation for $\sigma\tau$.
The phrase "degrees of freedom" refers to the number of independent real number valued parameters (as does the more formal term "dimension").
So, for example, each orientation preserving isometry of $n$-dimensional Euclidean space $E^n$ is a composition $T \circ O$ of a unique orthogonal transformation $O$ followed by a unique translation $T$, and these two factors are independent of each other. Translations are parameterized by vectors which have $n$ degrees of freedom, and rotations of $E^n$ are parameterized by orthogonal $n \times n$ matrices which have $\frac{(n-1)n}{2}$ degrees of freedom. Since the degrees of freedom of translations and of rotations are independent of each other, together they have $n + \frac{(n-1)n}{2}=\frac{n(n+1)}{2}$ degrees of freedom.
Now let's bring in reflections. Let me use $R_n$ to denote the reflection in the coordinate plane $x_n=0$. A general orientation reversing isometry can be expressed uniquely in the form $T \circ O \circ R_n$ where $T,O$ are independently chosen translation and orthogonal transformation. It follows that a general isometry can be written uniquely in the form $T \circ O \circ (R_n)^e$ where we independently choose three things: the translation $T$ with $n$ degrees of freedom; the orthogonal rotation $O$ with $\frac{(n-1)n}{2}$ degrees of freedom; and the exponent $e \in \{0,1\}$.
The choice of the exponent $e$ is discrete --- either $0$ or $1$ --- and this does not represent a "degree of freedom" because it is not a real number valued parameter.
By the way, what makes the calculations in my answer work correctly is the fact that the decomposition $T \circ O \circ (R_n)^e$ is uniquely determined by the isometry. In your question, where you use the fact that every isometry can be written as a composition of reflections, that composition is not unique, and so using it to count degrees of freedom can lead to errors.
Just as an example, picking any even number $n \ge 2$, every translation of the line can be written as a product of $n$ reflections. Since a reflection of the line has $1$ degree of freedom (namely, the reflection point), this would seem to be a proof that the translations of the line have dimension $n$. Since $n$ is arbitrary, there is an error in this argument... which I will leave you to ponder.
Best Answer
You can rotate at the intersection points of $4$ tiles by $180$ degrees. There are two kinds of intersections points depending on whether say the upper right is white or grey.