Is the "Moebius Stairway" Graph Already Known?

They are called quartic Möbius ladders.

They are one of the fundamental classes in Johnson & Thomas's classification of internally 4-connected graphs, and crop up in matroid theory for the same reason.

http://dx.doi.org/10.1006/jctb.2001.2089

I don't know if this family has a special name, but it is a simple type of circulant graph. Consider this way to draw it (for 9 vertices).

I don't know why you say it has only two hamiltonian cycles as it has many (82 in fact). Here's one beyond the two obvious ones: 0,1,5,4,3,2,6,7,8.

For $n=5,7,9,\ldots,37$, the number of hamiltonian cycles is 24,46,82,158,316,650,1364,2892,6170,13206,28314,60760,130446,280120, 601600,1292102,2775226,5960822.

Not in OEIS. Can you fit a formula or recurrence to it?

[Added] David Zhang has found an empirical recurrence for the numbers, which I'm sure is correct. Also, he is correct that I counted each cycle once in each direction -- I used a program designed for digraphs. I'll divide by 2 from now on. We can solve the recurrence with Maple's help. Let $\omega_1,\omega_2,\omega_3$ be the zeros of $x^3+2x^2+x-1$. Then the number of cycles for $n=2k+1$ (if the recurrence is correct) is $$ 2 + 2k + \sum_{j=1}^3 \frac{1}{(\omega_j+1) \omega_j^{k+1}}.$$ Approximate values are: $\omega_1=0.4655712319$, $\omega_2,\omega_3 = 1.232785616\pm 0.7925519925i$. Obviously the terms with $\omega_2$ and $\omega_3$ quickly become negligible. From $n=7$ onwards, the number of cycles is the nearest integer to $$ 2 + 2k+ \frac{1}{(\omega_1+1) \omega_1^{k+1}}.$$

I think Gerry's right; this is the construction that Kocay & Kreher in Graphs, Algorithms, and Optimization call the Möbius lattice (definition 13.18 on p. 365 of the 2004 edition, definition 15.21 on p. 403 of the 2016 edition, just after Möbius ladder). Their projective embedding of $L_7$ below corresponds to your graph (with appropriate vertex labels).

(The dotted line and $D_i$ are about changing this to a toroidal embedding.)

It doesn't seem like their name for this family of graphs is widely used; I found one undergraduate thesis mention it in an aside and some possibly relevant physics research.

As to further study, Möbius lattices come up in two exercises of Kocay & Kreher:

(1) Show that the Möbius ladder $L_{2n-2}$ is a minor of the Möbius lattice $L_{2n+1}$ for $n \ge 3$.

(2) Show that the Möbius lattice $L_{2n-1}$ has an embedding on the torus in which all faces are quadrilaterals.