I prefer a split. Though for this to make sense, both myself and my partner have to have a relatively similar sex drive. Right now we do not, so it's on her to initiate so I don't feel rejected so often.

We both initiate, we both have high sex drives so a lot of times the response is "I was just about to ask too". Our normal approach is to just walk up to the other one naked or pantsless. We also really like to catch each other masturbating, we both will masturbate when the other isn't home but we know they'll be there soon.

But being approached with no previous warning is a turn on for me

Same.

I think it's just not as interesting if only one person does all the initiation. I prefer her initiating but I end up doing it more myself.

My wife has a chronic illness that means that she's often not feeling up to sex. She feels bad about it, so when I initiate things and she has to say no, not only do I feel rejection (despite her no not being related to lack of interest), and she feels bad which can perpetuate the not feeling up to it part.

She knows that I'm down for anything at any time, so what's worked for us is I give her a back and leg rub most evenings, and if she wants to go further she spreads her legs a bit to give me permission to start foreplay.

I guess that technically means I initiate it, but only after getting permission to do so.