Real preciprocating engines, need an even number of strokes. You must ALways have at least the INTAKE, COMPRESSION, POWER, EXHAUST stokes.
They require a down, up, down up movement of the piston. You can combine the 2 upstrokes and 2 downstrokes to make a 2 stroke engine.
As you suggested of combining the power and exhaust phase, not only does that require the pison to be moving in opposite directions, but it also has an engine that producec no power what so ever.
After combustion you have high pressure gases, that push the piston down. If you opened the exhuast (had any gas removal system) then the high pressure gases would simply go straight out the exhaust as a path of least resistance.
All odd numbered stoke engines are all pseudostrokes, basically they are 2 or 4 strokes that are slightly modified.