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Does it really matter which is the primary and secondary?
Yes, it really does matter.
Or can I just connect the secondary to mains and get lower volts on the former primary coil?
Don't try it. You might live but the results won't be any good.
I know, I have to consider maybe the higher resistance of the thinner wire, but from my understanding it really shouldn't matter so I could use this chunky transformers
without modification to get lower voltage and high amps just by connecting it different way around.
2nd time: it does matter. The induction in the core will be all wrong if you excite it with the secondary winding whose number of turns is so at variance with the true primary. There is more than just the turns ratio to consider, you see. There is the mag circuit that actually transfers the energy between coils and the setpoint of flux in the core should be roughly what was originally designed. Also, the microwave transformers are gapped core so their loss behavior and other parameters are far from ideal.
I didn't had the luck to get my hands on one of these yet so I wasn't able to do any experiments
Do it like the others do: chisel out the secondary and wind your own. That way the mag circuit at least works the way it was supposed to.