caused

While the sentence you wrote is is technically correct, it would be more natural to say :
We apologize for any inconvenience this may have caused.

Evo25we apologise for any inconvenience caused.
This sentence above is right?

It's correct. We apologise for any inconvenience [that may have been] caused. Caused is a past participle.
CB

From Caris' point of view, it could be simple past. We apologize for any inconvenience [this] caused. (I'm not sure if that's a legal eliptical.)

Well, I suppose it could be passive, simple past: We apologize for any inconvenience [that was] caused. (assuming that when the participle is used to form a tense, it's called [part of] a verb ???)

- A.

It is the past participle.  You have reduced a relative clause to a passive participle phrase.
The relative clause would be which was caused.  

To reduce a relative clause to a participial phrase, you delete the subordinator and replace the verb with the past participle or active participle, depending on the voice of the relative clause.

You can only do this if the re

it is actually 'any convenience which this caused' but you have elipsed the relative subordinator which.  It looks like caused is the past tense (it is in the relative clause), but it is actually the past participle once the subject and relative subordinator are removed.  This is clearer in the following example:
We apologise for any food which was eaten. --> We apolog

Thanks, richard. I'm not sure if I'll live long enough to learn the correct terminology for these things. I guess I always assumed the predicate portion of a relative clause remained unscathed - that is, followed the same rules that would apply to an independent clause.

That is the vase which fell in the earthquake.

That vase fell in the ear

richard_s

richard_sIt is the past participle. You have reduced a relative clause to a passive participle phrase.
The relative clause would be which was caused.

To reduce a relative clause to a participial phrase, you delete the subordinator and replace the verb with the past participle or active participle,

The rules come with exceptions. Richard's explanation is true but "We apologize for any food eaten" = "We apologize for any eaten food," just as "We apologize for any rotten food."
doesn't seem to be covered under this blanket.

Both are past participles used passively.
Working with the undercover police, the store owner has recovered a lot of valuable merchandise stole

yes, participles can also be used as adjectives before a noun.  That is how get the old interested student vs interesting student problem.  In the case of interesting student, we are using the active participle as an adjective.  Interesting should not in this case be confused with the gerund (which looks the same).