In RSA, it is basically equivalent to use the private key $d$ to encrypt as it is to use the public key $e$ to encrypt. They are exactly inverses: $e=d^{-1}$.
I need to know, are there other public key schemes like this? E.g., I do not think Diffie-Hellman is like this.
What you seem to be looking for is a scheme like the following: It consists of two algorithms, a key generation algorithm $K$ and a "key use" algorithm $U$.
The key generation algorithm outputs a pair of keys $(k_0, k_1)$. The "key use" algorithm takes as input a key and an element from some set $S$ (which may depend on $k_0$ and $k_1$), and outputs an element of a set $S$, and satisfies $$U(k_b, U(k_{1-b}, s)) = s$$ for all key pairs $(k_0,k_1)$ output by $K$, all $b \in \{0,1\}$ and all $s \in S$.
Security-wise, for $b = 0$ and $b=1$, given a random $s \in S$ and the key $k_b$, it should be hard to find $t$ such that $U(k_b, t) = s$.
You can trivially turn this into a public key encryption scheme by choosing one of the keys as the encryption key and the other as the decryption key. Ditto for signature scheme.
To get secure schemes, you need to include padding schemes, hash functions or any of the usual stuff.
I don't off-hand know of any examples of such schemes, except for RSA with random exponents or RSA look-alikes. (RSA with small $e$ is not secure.) Diffie-Hellman is certainly not like this.
I don't know if this is a useful concept. It is somewhat similar to trapdoor one-way permutation, which you don't hear so much about anymore.
