On the Repubblica.it website today, there is a series of "foto carine", with the headline:
Oops, mi si è incastrato il cucciolo
I have never seen this type of construction with mi and si next to one another. I figure the meaning is something like "My puppy trapped himself." If that is the case, couldn't one say "si è incastrato il mio cucciolo?"
What sort of construction is this? Are there other similar constructions?
This is a quite usual construct in Italian: often, when in English you would use a possessive adjective (“I wash my face”), in Italian you use a possessive pronoun instead: mi lavo la faccia. Here, lavo la mia faccia would sound like something a foreigner (or perhaps Google Translate) would say. And where in English you shake someone's hand, in Italian gli stringi la mano.
Other common phrases of this kind include mi fa male il/la [parte del corpo] = “my [body part] hurts”; or le tengo i bambini = “I babysit her children”, and so on.
In your case, it is accidental that the verb is reflexive, so needing a further pronoun si. It could as well be, say, mi è caduto il cucciolo for “I dropped my puppy”.
mi è caduto il tuo cucciolo that can be translated as I dropped your puppy
A more strict (literal) translation could be:
the puppy trapped himself to me
This kind of construction is used to highlight the emotional or personal involvement of someone who isn't not the subject of the sentence.
Few other examples:
Sempre caro mi fu questo ermo colle - Always dear to me was this solitary hill (Leopardi)
Mi si è rotta la lavatrice - The washing machine is broken to me (this is not used, meaning it is wrong, but the literal translation would be written in this way, though you actually need to translate with my washing machine has broken.)
