As Marswill noted they have to be on opposite phases ... here's an image, grabbed off
this site, showing how the 120/240 volt stuff works in ordinary house setups:
What you would do in this case is connect a wire to the correct half of Plug A and the correct half of Plug B (as shown in the diagram) so that you get the 240 volts. If you use the wrong half of one plug, you get 120 volts, and if you use the wrong half of both, you get zero volts. Basically, you have to span the entire secondary winding, rather than one or the other center-tapped halves.
You'd still be pulling a continuous 13.75 amps on this setup, so with the 20% de-rate rule for house wiring you need two 120-volt "17.1875 amp" circuits (which really means 20 amp circuits like those used for kitchen wiring).
Typically that means hiring an electrician (most garage circuits are just 15 amp) so you might as well just have a dryer outlet installed and be done with it.