Say you have two servers with certificates proving their domains (e.g. for use with HTTPS).
Can one of these servers use their certificate as a client certificate when talking to the other, so that both servers can be certain of the other's domain name? E.g. "https://example.com" can get a request that it's certain is coming from "https://elsewhere.com", so can safely send secrets specific to that domain (even if it has never seen that domain before).
(EDIT: I'm aware that client authentication exists in general for TLS, but I'm particularly interested whether the same certificates from the "https" PKI are useable as client certificates. For instance, usage flags would prevent them from "just working" with existing implementations.)