The Big Lebowski and Something Wild. A pair of fish-out-of-water comedy with different kinds of wackiness.

Oh, there was a pairing I did once - SOMETHING WILD and INTO THE NIGHT (Jeff Goldblum and Michelle Pfeiffer). Very similar in the basic story line, in the way the movie swings wildly from comedy to thriller and back, and in the way that quirky characters and familiar faces keep popping up. The main difference being SW is a road movie, and ItN is an urban one.

I can think of a couple:

Happenstance and Amélie, two French films about chance/destiny and interconnected lives in Paris starring Audrey Tautou. They couldn't be more different in tone and style.

Novocaine and Little Shop of Horrors to see dueling portrayals of dentists by Steve Martin.

Maybe Double Indemnity and Body Heat for classic noir and neo noir?

Close Encounters of the Third Kind and War of the Worlds for two sides of Spielbergian aliens.

Any dream double features you would put together?

Glitter and Gigli

I still have fond memories of a great double feature at the Brattle: Buena Vista Social Club and Genghis Blues.

And when I was a pre-teen, I remember a great cheesy environmentally conscious horror pairing: Frogs and Godzilla vs. The Smog Monster.

Cheesy mutant animals on the prowl movies: Them! and Tremors would be good.

I have always wanted to do original film and homage double features:

I know Where I'm Going/Local Hero

Adventures of Robin Hood/The Court Jester

For those in the know: Hudson Hawk / Long Kiss Goodnight

Oh, and I think Jess already mentioned The Apple / Zardoz

Cheesy mutant animals on the prowl movies: Them! and Tremors would be good.

Them! was pretty good, with the giant ants and all.

I love Tremors. That would be a fun double feature.