Explain why negative reciprocals always have a product of -1?

A pair or reciprocals product is always 1 because a reciprocal is 1 divided by a number.

A negative times a positive is a negative.

Let a/b be the first slope. Let -b/a be the negative reciprocal. (We assume that a≠0 and b≠0).

The product is:
a .. -b ... -ab
-- x --- = ------ = -1
b ... a .... ab

That's pretty much the definition of a negative reciprocal.