The sentence "We have had the house painted" contains the
causative verb "HAVE" in its present perfect form. A causative
verb is used when the subject (one person) "activates" another
(second person) to do something. The subject does not perform
the action himself but rather "causes" another person to do it
for him. As in your sample sentence, the subject "WE" did not do
the painting themselves but they asked somebody else to do the
painting for them. Thus, "have had" in your sentence can be
labeled as follows: HAVE (Aux Verb - for present perfect) + HAD
(Main Verb - past form of "have") If we were to conjugate the
verb in the different tenses, it would appear as follows:
PRESENT: We HAVE the house painted every year. PAST: We HAD the
house painted last year. FUTURE: We WILL HAVE the house painted
next year. PRESENT PROG: We ARE HAVING the house painted right
now. PAST PROG: We WERE HAVING the house painted when it rained.
FUTURE PROG: We WILL BE HAVING the house painted next week.
PRESENT PERFECT: We HAVE HA