In this question, let's take the convention that $D_{2n}$ is the dihedral group of order $2n$.
It is natural to think of the cyclic group $C_n$ as acting on an $n$-gon via rotational symmetries. Similarly, $D_{2n}$ on an $n$-gon via rotational and reflectional symmetries.
Is there a similar way to interpret $\mathrm{Dic}_{n}$?
The presentation of $\mathrm{Dic}_{n}$ is
$$\left< a, x \mid a^{2n}=1,x^2=a^n, x^{-1}ax=a^{-1}\right>.$$ This seems awfully similar to the dihedral group in a lot of ways, in the dihedral group we also have a relation like $x^{-1}ax=a^{-1}$ to represent how reflection, rotation and reflection again corresponds to the inverse of that reflection. $a$ seems to behave similarly to the rotational element, and $x$ as the rotational element. Instead of $x^2=1$, we have $x^2=a^n$ which is the only rotation of order $2$.
Here's my thoughts - it acts on a $2n$-gon via rotations and also this kind of extension of a reflection, where the polygon is rotated in the third dimension by an angle of $\frac{\pi}{2}$. Applying the "extended reflection" twice isn't orientation preserving in the original plane, like a usual reflection.
I'm not quite sure if this is exactly right though and I'm struggling to visualise what's going on. Is there perhaps some other way to visualise this?