The word hydrostat reminds me of the classic routine of the Marx brothers viaduct for some reason. Looking at some of the musculature modeling done on the tongue it seems immensely complex but I have not seen much discussion between difference of the sides of the tongue. No one that I know is suggesting that there is a left and right tongue.
I am feeling that my left cerebral hemisphere image which controls what I call my right tongue is different and far more developed than my right cerebral hemisphere. If true playing my left tongue should be changing my right cerebral hemisphere.
What I feel with playing with my left tongue is that it cannot take it's full shape. I can proximate it but then it is not able move left and right freely the way it can in the right tongue mode.
My sense is my tongue is two hydostats side by side with a dominant side capable of much greater range of inhibition(relaxation) and contraction. The easier state for the left tongue/side is contraction but can do the necessary tasks for swallowing. Speech being a higher level function is mostly relegated to the dominant side.