How do the windings in a residential transformer create a neutral and why are the hots opposite phase?
The transformer doesn't manifest neutral: the ground rod does.



Or at least it should, according to electric codes. The "neutral" wiring is connected to the "safety ground" at exactly one point, and the safety ground is connected to this big copper rod stuck in the Earth. It's "neutral" because it's connected to Earth, and since you are probably also touching (perhaps indirectly) Earth, neutral won't shock you (absent any wiring faults, which is why there's a separate safety ground, which should be the same as neutral).

As a schematic, it looks like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Because the "neutral" is the center tap of the transformer, at any point the voltage at neutral will be exactly halfway between the voltages of the "hots". Thus, V1 is 120V, V2 is 120V, and if you add the two together you get 240V at V3. Or put another way, V1 = V2, both in terms of AC, or DC voltages at any instant.

The reason they are in opposite phase should be evident from the schematic: V1 has neutral on the "-" side, while V2 has it on the "+" side. As I've drawn the schematic, V1 and V2 are actually in phase, but if you flipped V2 around so that the "-" was on neutral, then they would be opposite phase. Because this is AC, which side is neutral and which is hot is largely irrelevant, except when you consider minor details like safety.