Public and private key in a key pair are distinct aspects of an underlying mathematical structure.
Let's take some examples:
In RSA, the public key consists in a big composite integer n (called the modulus), and a (usually short) odd integer (the "public exponent"). The private key is, ultimately, the knowledge of the prime factors of n. In practice, n = pq for two big primes p and q, and the private key is p and q.
In ECDSA, we operate over an elliptic curve, which is a weird mathematical animal that contains "points". A curve is a group of a certain size q. The private key x is a non-zero integer modulo q (that is, an integer between 1 and q-1); the public key is a point on the curve, which happens to be the result of applying the group law x-1 times on a conventional, fixed point (called the "generator").
In both cases, there is a strong mathematical relation between the private and public key, but they still are very distinct things. Also, the structure on which asymmetric cryptography works depends on the algorithm, and there is quite some variety.
Last but not least, DKIM is not about encryption. It is a mechanism that uses signatures. You apparently stumbled upon one of the many texts that purport to explain signatures as "encrypting the hash". Be aware that all these explanations are both confusing and wrong.