So I'm still trying to warp my head around the math behind gravitational waves (see what I did there?). They are odd beasts compared to the familiar EM variety; in particular they only have quadrupole and higher radiation characteristics because there is no gravitational analog to a negative-positive dipole.

When they propogate, they cause matter to oscillate in alternate axes. I made a little animation to understand what that might look like:

What seems to be missing here is a diagonal oscillation rather than purely in the X or Y axis. This turns out to be the quadrupole equivalent of polarization. This leads to the question, can you have circularly polarized gravitational waves by combining the linear X and + polarizations with a phase shift?

Turns out you can, and according to this paper by Smoot (sadly not the same Smoot) they look like a "twisted squashed tube":

Here's a hacky isometric projection where the ellipses are plotted at the same size and offset giving a 3D view: