Is there a way to alter the key data in a way that the algorithms for en/decrypting do it with the "reverse" key? I tried to swap e and d (because that is what it seemed to me from wikipedia the difference is) - for "signing" it worked (but I assume it was undecryptable garbage)... for "verifying" I tried to calculate d mod (p-1), d mod (q-1) and 1/q mod p with e instead of e for usage of the decryption algorithm for verification... but is it really what I think it is? If it was, I suppose it wouldn't have to be stored in the private key, because it would be very easy to calculate, having d, p and q...?
The point: If there is a way to feed a modified private key into the encryption algorighm for signing and the same for the counterpart - how do I do this? What modifications have to be done to the keys?