It's hard to make a list of verbs that behave the same as others. It's only by reading and being exposed to language that one can know what verbs admit certain constructions.
Instead, I have made some notes on the use of verbs ending with "-se". I hope you'll find them useful.
There are three main functions of "se":
1) Pronominal
a. Used when IO is preceded by DO:
· Les di un regalo. -> Se lo di.
I gave them a present. -> I gave it to them.
· El abuelo le leyó un cuento. -> Se lo leyó.
Grandfather read a story to him. -> He read it to him.
· Les abrieron la puerta. -> Se la abrieron.
They opened the door for them. -> They opened it for them.
· Le canté una canción. -> Se la canté.
I sang a song to her. -> I sang it to her.
b. Reflexive "se" - (Particular case of pronominals) Used when the subject performs an action on him/herself; when it's a reciprocal action between two or more individuals:
· Juan se peina.
Juan combs his hair.
· Los niños se lavaron las manos antes de comer.
The children washed their hands before eating.
· Te levantas muy temprano todos los días, ¿nunca descansas?.
You wake up very early every day, don't you ever have rest?
· Me dije (a mí mismo) que tenía que decir la verdad.
I told myself that I had to tell the truth.
· Mi esposo y yo nos conocimos en la universidad.
My husband and I met in the university.
· En este foro nos apoyamos unos a otros para aprender idiomas.
In this forum we help each other to learn languages.
*There are some verbs that are exclusively pronominal, like "arrepentirse", "quejarse", "adueñarse", etc.
· Te vas a arrepentir de haberme insultado.
· Pedro siempre se está quejando de su mala suerte.
· La nueva empresa se adueñó de todo el mercado.
And there are some verbs that take a nuance of meaning when expressed as pronominals, like "irse", "dormirse", "salirse", etc.
· Dormí en la casa de mi abuelo. - Me dormí en la casa de mi abuelo.
I slept at grandfathers place. - I fell asleep at grandfather's place.
· El jefe salió de la junta. - El jefe se salió de la junta.
The boss went out from the meeting. - The boss left the meeting.
(I think this could be the answer to your question, Kitten)
c. "Voz media": it expresses an action that happens to the subject, without an apparent cause or when the cause doesn't matter (mostly because it's beyond anyone's control.)
· La casa se incendió.
The house caught fire.
· Las calles se inundaron.
The streets got flooded.
When the verb is built as a pronominal and is combined with impersonal "se", it underlines the action happening on the subject (with no one having control over it, or the subject performing the action is irrelevant).
· Se me rompió la falda.
My skirt got torn.
· Se nos cayó el techo encima.
The ceiling fell upon us.
· Se me ocurre que podemos hacer un buen negocio.
It occurs to me that we can make a good business.
2) Impersonal sentences (where there is no subject)
· Se siente bien estar en casa.
It feels good to be at home.
· Se vive mejor en el campo que en la ciudad.
It's better to live in the country than in the city.
· Aquí no se discrimina a nadie.
Nobody is discriminated here.
· En estos días sólo se habla de crisis económica.
These days all people talk about is economic crisis.
3) Voz pasiva refleja
· Se habla español.
Spanish spoken.
· Se venden frutas frescas.
Fresh fruits for sale.
· Esa computadora se hizo en México.
That computer was made in Mexico.