Is the reciprocal of a product the product of the two reciprocals

Well a reciprocal is the opposite of a number, so the if you multiply the opposite of a number twice, it will equal the original number..
Unless you mean multiply the original by the reciprocal, then it'll equal a negative number.

After reading this like 10 times, I think it means:

\frac{x}{y}\ \cdot\ \frac{a}{b}\ =\ \frac{xa}{yb}

\frac{y}{x}\ \cdot\ \frac{b}{a}\ =\ \frac{yb}{xa}

In which case, yes.