I'd say it's generally a power move, even if you're the one that's getting dumped. Though it is better to be the one saying 'I'm done'. It's not so much the act of who does the dumping . It comes down to how you handle yourself during and after the breakup. If you suck up to her, or cry, trying to fix things - the respect just shoots down for her. She didn't come to you trying to work things out, or resolve things. She wanted to end it - so the best thing you can do is go 'Ok', and leave her on her own.
Like Serenity said, I have a similar view towards 'She's not in, I'm not in'. I used to spend a lot of time in my younger days dissecting a girl's actions. 'What did she mean by this?', 'Maybe she's upset about this', and other in-depth analysis. Eventually I realized that when it comes down to a situation like this, either a girl is working on your side with you, or she's working against you. Sometimes you can adjust for momentary stresses, but I think you can decide when that applies.
Here's a thought experiment you can go through. You're dating this girl, and for whatever reason, it's not working out. She's gotten annoying, clingy, neurotic, or you think she's not long-term relationship material - whichever. You break up with her. In one option, she breaks down crying, 'Why are you breaking up with me?', throwing a fit. Calls you horrible names, tries to put you down. The break up has gone from 'Alright, let's break up' to 'Geez this person is nuts'. She shows that she valued you to such a degree that she's lost control, embarrassed herself, and gone to extremes to try and bring you down. Showing the emotional depth it reaches to. It gives you all the further justification you needed for breaking up, and if there was any future pangs of 'What if you two stayed together, or got back together in the future?' - you just have to think of that last moment. Especially as the final memory of that person.
Alternate story, she goes 'Ok, let's break up', and it's relatively painless. She goes her way, you go yours. She shows that she's not going to degrade herself to desperately save a relationship that's done. You wanted it done - well, it's done, and that's ok. Possibly you may respect her in how she handled the ending steps of the relationship, and possibly that may open something in the future (if you wanted). You know that she won't turn into a crazy person, and thus it's safer to re-engage her.
This is not guaranteed, as there might be a strong underlying reason for the break up. And you might never meet or talk again. But the ending and how it's handled is a factor.
Overall, my opinion is that how you handle yourself as the breakup occurs and after is far more important than who decided it. Yes, having her get to the point of her deciding 'I don't want to be with this guy anymore' is a negative, but not making a fool of yourself in trying to save it, may raise it to 'huh, maybe I made a mistake...'. I'd go to even say that handling it smoothly brings up thoughts of 'Why weren't they upset? Why didn't they try to save it? Was I not worth it?'. I think it turns it from an initial one-sided dumping, to a mutual agreement. That's where you can draw the 'power' from.