In an excellent post several years ago, we learn that the period of the decimal expansion of a rational number $\frac{p}{q}$ must divide the multiplicative order of $10\pmod q$ assuming that there are no factors of $2$ or $5$ in $q$.
Length of period of decimal expansion of a fraction
How does the period of the decimal expansion of $\frac{q}{p}$ relate to that of $\frac{p}{q}$? Let us assume that $p, q$ are coprime integers. (I have added this in response to an answer given.)
I'm not sure how to prove this, but it seems like the periods of $\frac1q$ and $\frac pq$ are identical. So what you are asking is if for any primes $p$ $q$ how the periods of $\frac1p$ and $\frac1q$ are related. I would say they are not.