First I am curious about the 6 phase inverter you are referring to - and what is the application.
As for the 180 Phase shift - that is how a 6 phase system would look, relative to a ground point. The only real way to discuss is to look at the vector diagrams. ( http://electrical-engineering-portal.com/understanding-vector-group-transformer-1 )
For the Rectifier - a standard 3 phase system will be called a 6 pulse, meaning as each diode begins to conduct you see a current spike, with 60 Hz for each phase you see 2nd harmonic - on the DC side you see this effect from all three phases to there is 360hz ripple - etc. Both the AC and DC issues need to be dealt with. - Sorry to not write the whole thing up there are lots of good resources covering this. (
Example) .
As for feeding a motor I mentioned direct feed ( no inverter) - also depends on the motor and application if there is a benefit. In general the main reason I see a transformer put between an inverter and motor is to use Low Voltage inverter to drive a Medium Voltage motor. - Still not 6 phase. Most of the 6 phase inverter / motor combos are I am finding are academic - again they can be done, and have some technical benefit - but in the end they are not practical - typically due to complexity or cost.
Last point - as I said a different set of issues... 12 pulse rectifier is to reduce harmonics seen by the source. (consider the rectifier as being completely independet of the inverter)--- The inverter (feeding a motor, or a computer ( like a UPS) - or other devices like the Grid in a Solar Inverter - want to generate a clean signal for what ever the load is ( motors are actually about the easiest case - but they do not like the high dV/dt "noise" from the switching - when VFDs first came on the market they often needed ruggedized ( expensive) motors that were "Inverter Grade" to deal with this. MOSFETs are very close to an ideal switch and generat little losses on each turn on/ off - so many designers get used to running very high switching frequensies - IGBTs have much greater switching losses but can handle higher voltages and currens ( they parallel a little easier as well) - so for larger power the Switching frequency choice a major design consideration.