I'm interested in reading more about this grammatical phenomenon and its rules but I don't know what to call it. Is it considered a tense, or does it have a different term? There's no corresponding phenomenon in French or English, e.g. no "tohaveus" or "avoirnous", so I haven't encountered it before.
A verb that requires a reflexive pronoun is called a verbo pronominal. This pronoun is attached at the end of the verb in the infinitive, gerund and (affirmative) imperative tenses, but appears before the verb and separate from it in other tenses1:
- Vamos a cansarnos mucho.
- Está duchándose.
- Se cayó al suelo.
- No te rindas.
A pronoun that is attached at the end of a verb is called a pronombre enclítico. This may be a reflexive pronoun (that refers to the subject), but it may also be a pronoun that refers to an object different from the subject:
No quiere moverse de allí (reflexive enclitic pronoun).
No necesito mi bicicleta antigua. Voy a venderla en eBay (non-reflexive enclitic pronoun "la", that refers to the feminine object "la bicicleta").
1In old literary texts, personal tenses with attached enclitic pronouns such as the famous "Érase una vez" or "díjole" are common. This is not common anymore.