Can I install 120v and 240v receptacles behind the same two-pole breaker?
What you built is called a multi-wire branch circuit.
You must follow all the rules for a multi-wire branch circuit, including pigtailing neutral and not splicing neutral through a device (e.g. no using the 2 silver screws on a recep).
If it has only 120V loads, it can use breakers which do not have common trip (e.g. handle tied breakers or quadplexes without common trip).  Yes, you are allowed to mix both  120V and 240V loads without restriction... however, if you do, the breaker must be common trip.
In fact, they make a special receptacle just for that.  It has a NEMA 6-20 socket and a NEMA 5-20 socket in the standard receptacle package.  They also make one with a 6-15 and 5-15.