I understand the concept of in and out inertia energy, i.e. the energy it take to accelerate the piston/rod mass on the power stroke at TDC and the returning of this kinetic energy while stopping the piston/rod mass at BDC. I know that this area is not commonly looked at as a big area of loses, but there has to be something there.
Your question suggests that perhaps you don't understand the "inertia energy" as well as you think you do. If you accelerate the piston up and down continuously in a frictionless engine, regardless of the acceleration rates up vs down, etc., whatever you put into the piston will have come out completely when it comes to a stop (top or bottom). The fact that there is a varying rate at which you put the energy in or take it out doesn't make a difference- the stopped piston hasn't accumulated any energy (where would it store it?).
If you consider friction, hysteresis, etc., then you can have energy losses, and the piston can get a little hotter.