this is a strange cat but its still very cute

They do not, that is an upset stressed out alligator.  Alligators will roll their eyes back and close them in defense and out of submission when they feel they are in danger of being hurt.

This alligator does not feel safe and is rolling its eyes back in defense, and is displaying signs of stress.

This is not cute.  This is not a happy animal.

Many people do not know that reptilians over all do not operate like dogs or cats and show stress and pleasure in completely different ways, please pass this around to help get the word out and hopefully save some future scaley friends of ours from stress like this that can be incredibly bad for their health.