I've got mixed and matched cheap tires, because......well.....I'm cheap.

195 / 65 R14
Front: Douglas XTRA-TRAC (at the moment)
Back: BF Goodrich Excentia Touring (at the moment)
I've covered a lot of ground with a lot of cars with different types of tires, and never had a problem.
This car with any tires has had this problem as long as I've owned it (since 97 I think). It does fine until I hit a patch of pavement with rain grooves, then all of a sudden it is like the road starts telling my car where to go. Tossing it back and forth.
I will switch to better tires all the way around if it will fix the problem, but it seems strange that I would have to.