Mirror shots are another favorite of Tarantino's. We get two of them in Reservoir Dogs. Just before Orange leaves to meet Joe about the job, he looks in the mirror, checks himself out, and tells himself not to "pussy out"—he won't get hurt, he'll be okay, and they'll believe every word he says, etc.
Mr. Orange looks like he's trying hard to convince himself he can pull this off. He's assuming a new identity and it's an intimate moment with the mirror. Tarantino's not going to have Orange go into therapy and reflect (pun intended) on his fears and insecurities about the job, but he doesn't have to. The mirror scene does it.
Same with White, who shows up and spends some serious time looking in the mirror combing his hair. He's trying to pull himself together after the bloody getaway, but he seems to be spending more time than is absolutely necessary in front of the mirror.
Our guess? He's reflecting (sorry again) on what's happened and struggling to put it in order. He's agitated and confused; maybe looking in the mirror helps him orient himself to where he is and who he is and what he has to think about.