Symmetric keys are only to be used in a peer-to-peer way so it would be pointless for the receiver to modify JWTs for which only he and the sender have a shared key (and he is the intended recipient).
Asymmetric key signatures (in JWTs as well as in general) are produced by the sender with the private key and verified by the receiver with the public key. The consumer/receiver is given only the public key which happens out_of_band (i.e. through another means of communication than the one you use to exchange the secured data).