Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. More generally, if $\{Y_n\}$ is an Omega spectrum $X=\dots \times Y_1 \times Y_0 \times \Omega Y_0 \times \dots$ is an example.

I ask because I have a space (related to stable block bundles) which does not seem like it should be contractible but appears to be its own loop space.