"Third conditional" or "conditional III" is a pattern used to refer to hypothetical situations in a past time frame, generally counterfactual (or at least presented as counterfactual, or likely to be counterfactual). Here the condition clause is in the past perfect, and the consequence is expressed using the conditional perfect.
The condition clause can undergo inversion, with omission of the conjunction:
Here are some more examples:
Another possible pattern (similar to that mentioned under the second conditional) is if it hadn't been for... (inverted form: had it not been for ...), which means something like "in the absence of ...", with past reference.
For the possible use of would in the condition clause. In the main clause, the auxiliary would can be replaced by could, should or might, as described for the second conditional.