###You are OK, got you first time with dig as follows:
[all@canopus all]$ dig hardell.mine.nu mx
 
; <<>> DiG 9.2.2 <<>> hardell.mine.nu mx
;; global options:  printcmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 20942
;; flags: qr rd ra; QUERY: 1, ANSWER: 1, AUTHORITY: 0, ADDITIONAL: 0
 
;; QUESTION SECTION:
;hardell.mine.nu.               IN      MX
 
;; ANSWER SECTION:
hardell.mine.nu.        43200   IN      MX     10 hardell.mine.nu.
;; Query time: 293 msec
;; SERVER: 192.168.0.3#53(192.168.0.3)
;; WHEN: Sun Feb 22 22:13:03 2004
;; MSG SIZE  rcvd: 49
 
###Going the other way around:
; <<>> DiG 9.2.2 <<>> mail.hardell.mine.nu
;; global options:  printcmd
;; Got answer:
;; ->>HEADER<<- opcode: QUERY, status: NOERROR, id: 16414
;; flags: qr rd ra; QUERY: 1, ANSWER: 2, AUTHORITY: 0, ADDITIONAL: 0
 
;; QUESTION SECTION:
;mail.hardell.mine.nu.          IN      A
 
;; ANSWER SECTION:
mail.hardell.mine.nu.   43200   IN      CNAME   hardell.mine.nu.
hardell.mine.nu.        60      IN      A       165.165.84.239
 
;; Query time: 496 msec
;; SERVER: 192.168.0.3#53(192.168.0.3)
;; WHEN: Sun Feb 22 22:18:36 2004
;; MSG SIZE  rcvd: 68
All appears OK