Message signature using RSA in my textbook it is stated, that if we have a message, it's signature using RSA and modulo, we can also sign the inverse of the message. 

However, it is not explained why can we do so.

Could someone explain to me the reasoning?
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.