Why is a MAC verification algorithm required when we can simply create a new MAC and compare the received one with the one calculated?
Signing and signature verification are concepts related to public key cryptography.
In case of message authentication codes, there is a single secret key and one function:
T = MAC ( K, M )

Where:

T - tag
MAC - MAC function
K - secret key
M - message

Messages can be signed by either the client or the server and verified by the other, depending on the case. Both parties calculate MAC the same way, using the same function and secret key, over the same message. There is no separate function for MAC verification. The only extra step on the receiving end is associated with comparing if both received and calculated tags match, and then acting accordingly. But this has nothing to do with a MAC algorithm itself.