Thanks to Lane for recommending a fantastic band: The Fiery Furnaces. It's rather late for me to go about trying to describe them. I'm horrible at it anyway.
I initially got a Mates of State vibe from them. There's piano there. Which is like an organ, I guess. And guy-girl vocals. But I just chalked that up to hearing them about the same time I saw Mates live. Turns out it's a brother-sister duo, so to some extent, I was at least somewhat close on that initial impression.
I also detect a hint of The White Stripes in there. Not exactly, but sort of. They've got that sort of bluesy, roots feel at time. The instrumentation is much wackier, and definitely not as sparse as the Stripes (or Mates of State for that matter). In any event, AMG lists the Stripes as a "similar artist," so again I am vindicated.
All in all, this is great stuff. The tracks "Two Fat Feet" and "Asthma Attack" are my favorites of the generally-quite-good album Gallowsbird's Bark. Check it out.