At D or G# the piano is a mirror. Take D or G# as a first note you can play your hands symmetrically. Meaning if your right thumb plays A your left thumb plays G, if your right index finger plays Bb your left index finger plays F#. There are a couple engaging experiences that follow.
First of all whatever your dominant hand, you simply command the weaker hand to follow. Not the same effort as thinking about separate directions for left and right, playing symmetrically means one just has to follow – it’s how our bodies are designed – a nice byproduct is that the weaker hand, usually the left, will achieve more velocity but it works the other way too if the pianist is a lefty then their right hand will experience more freedom to play swiftly.
The other attractive aspect of playing this way is whatever harmony you know goes out the window and that’s the place I personally enjoy the most. Trying to improvise with this set of conditions and not sound like an exercise.