what type of binary can be detected by observing the back-and-forth Doppler shifts of its combined spectrum?

Answer 1

Spectroscopic.

Answer 2

Hi. Spectroscopic doubles, as answer 1 points out. The really interesting stars are multiple doubles. Castor, for example is a close (visually) double with a third star orbiting them. A spectroscope reveals the EACH of the three stars is a double, so Castor is actually six stars. Neat! http://www.astro.uiuc.edu/~kaler/sow/castor.html