Message signature using RSA
Let the modulus is $n$, the message $1&lt; m &lt; n$. In your textbook RSA the signature is $$s = m^{d} \bmod n,$$ where the $d$ is the private key.

Let the target signature be $s' = s^{-1} \bmod n$ and $m' = m^{-1}$

Let expand $s' = s^{-1} =  (m^{d})^{-1} = (m^{-1})^{d}$ 

Therefore $s'$ is the signature of $m'$.

That is one of the many other reasons that we never use textbook RSA in practice. We should you RSA-PSS or RSA-FDH for signatures.