Back in the '50s this was tried .... it was called a Swash Plate design . Fine for clean conditions like AC compressors but otherwise it was a dud .
Just because it can be done doesn't mean it's worth doing .
+1 with what Bruno said.
Remember: unlike sealed A/C compressor, which runs in a sealed, uncontaminated environment lubricated with compressor oil, etc. (which remains uncontaminated/not deteriorated) for a long, long time; exposing to fuel and air, and dirt in air (abrasion), combustion byproducts, blowby, moisture/condensation, etc. there will be no guarantee on how long/reliable such design would be, and also the inherent energy conversion efficiency also.
Just because it looks good in theory doesn't mean it would be meaningful/practical in real-life situation.
Afterall: we have been constantly refining/improving the venerable internal reciprocating combustion engine design for a century now, it would be challenging to start from scratch to design something even better...
I know it's not 100% impossible, but it would be challenging.