Yes. Ownership is the legal bit and nothing to do with the mortgage. Pretty sure the mortgage can still be in both your names too (cue loud sound of sucking eggs...)
Are you moving or just super organised about your finances?
We are waiting to move into a new house, the mortgage is in dh's name only as my parents are moving down here shortly and their house is going in my name. But we have some legality forms that state I am jointly entitled to live in our new house (can't remember all the legal terms for it, but basically dh can't just chuck me out! )
Our circumstances are similar. DP is self employed and doesn't really have any income to speak of, so we based our mortgage on my salary. To get DP's name on the mortgage we had to say that DP had some income, but we put down that he earned around £500 a year and the bank were happy with that. We didn't even have to provide proof.
As far as owning half the house you need to speak to the solicitor who writes out the deeds and make sure that's what is written.
I never really shop around - perhaps I should - but the only thing I really care about is complete flexibility - the one time I went for a fix I ended up paying penalties when I paid off a lump sum!
But yes I can see the attraction at the moment (ponders very large new mortgage acquired at the top of the housing market...)