No inexpensive way to do this safely or reliably. (See user71659's answer for an approved device, but the cost and "no GFCI/AFCI" constraints are important.)
The reason 240V -> 120V works is the principle of the Multi-Wire Branch Circuit. This is a 240V circuit that uses one neutral together with two 120V hot wires that are on different "legs". That means there is 240V between the two hots, so you can power either 240V or 2 x 120V or both from the same circuit.
There are a number of safeguards involved, particularly to make sure that (a) if you turn off one 120V hot wire in the breaker panel that you turn off the other at the same time and (b) that the two 120V hot wires are on different legs and not on the same leg.
While using a crazy adapter is not the normal way to create a 120V receptacle on a 240V circuit, it is not inherently wrong. There is a current overload protection issue - e.g., using a 30A 240V receptacle to power a 15A or 20A 120V receptacle requires providing 15A or 20A overcurrent protection for the 120V receptacle, but that can be done. The cheap adapter cables are not UL or ETL listed because they don't provide this protection. However, this GE adapter from Home Depot:
does provide protection - a circuit breaker on each 120V socket.
However, going the other direction you have to make 100% certain that the two "source" receptacles are on opposite legs. That is easier said than done. MWBCs are relatively common in kitchens, but even then they could be configured for top/bottom in each duplex receptacle (in which case this type of adapter would work well) or for alternating duplex receptacles (i.e., the MWBC splits in the first receptacle box), which has the advantage of allowing for GFCI/receptacles to provide GFCI protection.
You are really much better off running a new 240V 20A circuit and installing a standard NEMA 6-20 receptacle. Then you can put a 6-20 plug on the end of the appliance cord and you're all set. But remember, this will need GFCI protection, which means a GFCI/breaker as GFCI/receptacles are only generally available for 120V circuits.