Make your life easier. Don't rewire the motor to be more like a DC motor or something. Just reverse the motor the way universal motors are normally reversed.
Here's one of my sewing machine motors:
Connect it to a double pole, double throw (DPDT) relay like this:
simulate this circuit – Schematic created using CircuitLab
You can use an electromechanical relay (one with a coil in it) or a solid state relay (no coil, all electronic.)
In either case, you will need a relay with contacts rated for 240VAC and more than 1A.
The rating for the switching side of the relay will depend on the circuit that will control the motor. If you are using an Arduino (or other 5V microprocessor,) then you'd want a relay that can be driven be 5V and (very) low current.
The electronics will have to go outside the motor housing. I don't think you'll find a suitable relay that will fit inside.
Since you are playing with 240VAC, make sure to properly insulate your wiring and use grommets where the wires pass through the metal. Be sure to insulate the switching circuit and provide isolation for the switching side (whatever is driving the relay.)
Rewiring the motor to put the coils in parallel is problematic. The coils were designed with the understanding that the coils will be in series, so each only has half of the 220VAC across it. Putting them in parallel exposes each to the full 220VAC. Because of the higher voltage, more current will flow through each coil when they are in parallel. That's probably a bad thing.
Be very careful when working on the insides of a sewing machine motor. I just learned the hard way that the windings in some of them use aluminum wire. If you break a connection, you cannot solder it back together with normal solder.
Aluminum breaks easier than copper and can't be soldered easily. I tried, but couldn't do it. The connections inside the motor are crimped rather than soldered. The blue wires in the picture can be soldered, but not the motor windings.