Geometric argument why rotation by 180 degrees commutes with reflections I have no trouble determining the center of the dihedral group $D_n$ using an algebraic argument ($R_{180}$ is self-inverse). 
But I've been trying (without success) to find the geometric explanation why rotation by $180$ degrees commutes with any reflection. 

Is there a geometric argument why rotation by $180$ commutes with any
  reflection in $D_n$?

I have observed that this rotation maps each reflection axis to itself but I don't see how to proceed from there (if it's even possible).
 A: You're almost there. :) It may help to show that reflections across orthogonal lines commute: If $R_{1}$ and $R_{2}$ are reflections in orthogonal lines through a point $p$, then $R_{1} R_{2} = H_{p}$, the half-turn about $p$, and the same argument applies to $R_{2} R_{1}$.
Now if $R$ is an arbitrary reflection across a line through the origin, let $R^{\perp}$ denote reflection about the orthogonal line through the origin, decompose the half-turn about the origin as $H_{0} = RR^{\perp}$, and observe that each factor commutes with $R$.
(The algebraic version, of course, is that $H_{0}$ is induced by the scalar matrix $-I$, which commutes with every linear transformation, particularly with every linear reflection.)
A: Try labeling the vertices of the regular $n$-gon, $\{1,...,n\}$ and the observe the action of the two group operations on any vertex, in both orders. See if the vertex mapping is the same. 
For a start, you can try using choosing the reflection to occur about the line of symmetry passing through vertex $1$, and then noting that $1$ maps to itself, $2$ maps to $n-1$, $3$ maps to $n-2$, and so on.
