You are asking, can both public and private exponent be the same? It might work, but system collapses on itself, because in real life it would be useless. As we are sending pub key via the web. Anyone can intercept and decrypt the message with the same key.

Yes, math behind this work. But it seems that you are asking, can both public and private exponent be the same value? Answer is no, because you jeopardize the whole system. By choosing equal exponents, you create two identical keys. And if some eavesdropper steals public key, he can decrypt message.