Here's an interesting graph (from Wikipedia):
It looks like up through WWII the perception that Democrats spend and Republicans cut back was fairly accurate, but after that it kind of breaks down.
*Edit* - To be clear that's debt and not spending, and the debt can also be lowered by increasing revenue rather than decreasing spending.
Another trend I'm noticing is that it seems that during good economic times (20's, 50's-60's, 90's) the debt tends to come down, and during bad times (30's, now) and major wars (WWI, WWII) it tends to go up. There is that 80's increase that seems to go against that though.