I wonder if anyone has counted these curves, either exactly or asymptotically?
Let $S_n$ be an $n \times n$ subset of $\mathbb{Z}^2$ consisting of $n^2$ lattice points: a lattice square. Define a rectilinear filler curve for $S_n$ to be a simple closed curve that passes through each of the $n^2$ lattice points, and is composed entirely of vertical and horizontal edges. So the curve is what is called a "rectilinear" or "orthogonal" polygon in the literature. Every turn of such a curve is $\pm 90^\circ$.
I'd like to know the number $f(n)$ of distinct filler curves for $S_n$, distinct up to rotations and reflections. So if $C_1$ can be rotated and/or reflected to lay on top of $C_2$, then $C_1$ and $C_2$ are not distinct.
$f(2) = 1$, and $f(4) = 2$:
$f(n)=0$ when $n$ is odd, as can be seen as follows. View a filler curve $C$ as composed of unit-length segments connecting lattice points; call these the edges of $C$ (so two incident edges can be collinear). Each horizontal line $y = m + \frac{1}{2}$ for $m$ an integer crosses an even number of edges of $C$; similarly for vertical lines. So the total number of edges $E$ of $C$ is even. In Euler's relation $V-E+F=2$, $F=2$ (interior & exterior of $C$). So $V=E$. So $V$ must be even. But $V=n^2$ for $n$ odd is odd.
Already I don't know what is $f(6)$. It is easy to see the growth of $f$ is exponential in $n$, but I don't know more. In particular, I do not see how to recursively connect $f(n)$ to $f(n-2)$.