When washing your own body parts, it is appropriate to use the reflexive version of the verb: lavarsi instead of lavare, or often simply omit the specification. It's not very usual to wash someone else's body parts.
Lavarsi le mani, "Lavati i capelli!", Mi lavai i denti
or (less frequently)
Lavare le mani, "Lava i capelli!", Lavai i denti
You can always use the active form and specify whose body parts you are washing
Lavare le proprie mani, "Lava i tuoi capelli!", Lavai i miei denti
However, since the version without the adjective is very common, this version sounds overwrought and is seldom used unless you are washing someone else's body parts.
Lavare le mani di qualcuno, "Lava i suoi capelli!", Lavai i suoi denti
In which case you can also use the form with the joined complement, like so
Lavarle le mani, "Lavagli i capelli!", Le lavai i denti
However you can not specify both: the presence of the reflexive form of the verb and the adjective are redundant and are not correct Italian.
Lavarsi le proprie mani, "Lavale i suoi capelli!", Mi lavai i miei denti
mi lavo le mani
there is no reflexive verb.Mi
is complemento di termine and this construct is called forma riflessiva apparente ("seemingly reflexive verbal form").Mi lavo le mani
is structurally more similar tomi compro la macchina
than tolavo le mie mani
. Anyway I appreciate this doesn't answer your question. I don't know whylavo le mie mani
just sounds wrong. What is really weird is that sayinglavo le mie mani
kind of implies to my ears that you wash your hands as opposed to somebody else's.