1

Spanish non-finite or impersonal forms are: infinitive or infinitivo (CANTAR), gerund or gerundio (CANTANDO) and past participle or participio (CANTADO).
Then we have a number of tenses where the verb itself is conjugated: CANTO, CANTABA, CANTARÉ, etc. These are called simple tenses or tiempos simples.
Then we have tenses where an auxiliar verb haber is conjugated instead, followed by the participle of the original verb: HE CANTADO, HABÍA CANTADO, HABRÉ CANTADO, etc. These are called compound tenses or tiempos compuestos.

Then we have other forms where the auxiliar verb haber itself uses an impersonal form, and is followed by the original verb's participle: HABER CANTADO, HABIENDO CANTADO.

Questions:

  • Do these constructions have a name?
    Searching for haber infinitive I found Using the verb Haber in its infinitive form which explains how forms like HABER CANTADO are used, but it doesn't mention a specific name for them.
  • Does something like HABIDO CANTADO even exist? how would it be (or was) used?
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If you take the conjugation of the verb temer, or in this article about modelos de conjugación verbal you can find that the mentioned forms are called "formas no personales compuestas".

So basically you have infinitivo simple which is usually called just infinitivo:

temer

infinitivo compuesto:

haber temido

gerundio simple:

temiendo

gerundio compuesto:

habiendo temido

and finally the participio

temido

which doesn't have a compound equivalent.

I also found this article in spanish and english.

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