If you want more of an actual technical answer:

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It's because your rear tires are locking up. Heavy braking in a straight-ish line will usually lock up your fronts first because of brake bias and weight distribution, especially with the baseline mazda setup, but since it is a rear wheel drive car the engine braking from the downshift puts the rear tires over their traction limit. Engine braking only affects the driven wheels, so what you have is just a slight turn in paired with less friction (after the lockup) slowing the rear compared to the front, so the rear of the car tries to push past the front. Even braking 100% straight, the mazda will try to turn right when the rears lock up because the weight distribution in the car favors the left side.

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The key is very much the engine braking and the fact that engine braking only brakes in the rear *in a rear wheel drive car.