The definition of "reflex" is an action that is not voluntary - ie, not mediated by cognition.  From [Wikipedia][1]:

> ... sensory neurons do not pass directly into the brain, but synapse
> in the spinal cord. This characteristic allows reflex actions to occur
> relatively quickly by activating spinal motor neurons without the
> delay of routing signals through the brain ...

So asking if there are any mental or cognitive reflexes is like asking if there are any non-reflex reflexes.

Perhaps you are more interested in other types of involuntary processes that do go through the brain, but that would be a different question, and I recommend being careful about the definition, because the entire field of "perception" (just as a simple example) could be considered "involuntary" processing depending on the definition used.


  [1]: https://en.wikipedia.org/wiki/Reflex_arc