I've heard of partners checking others phones, and I find it deplorable. It's a breach of trust. Especially if one is doing it behind the others back. A phone, like a computer or a diary, is a collection of private thoughts, feelings, and conversations one has with others or themselves (in the case of the diary). We don't always want our thoughts to be known to everyone. We each deserve some privacy.
To me, the act of even asking to go through ones phone would mean that she doesn't trust me. Trust, at least to me, is the foundation upon one builds a relationship. If it's not there, the relationship is doomed to fail. I'd be angry she asked to do so. That being said, I'd be *furious* if she did so behind my back and I found out about it. That, to me, would be the death knell for the relationship. Not only would she not be trusting me, but she'd care nothing for my privacy, and attempting to do it secretly is akin to lying.
I'd personally never ask to see ones phone. She'd be allowed to have her own life and conversations. Also think on the fact that any conversations or such, generally are understood to be private by the person you were talking to. Maybe a coworker calls or texts needing to vent about something confidential, or a family member calls telling you of their medical issues. If someone goes through your phone, they now may know those details. They'd probably be mad at whomevers phone it was.
To summarize, I don't think it's right, and would not condone it.