We present Spitzer Space Telescope observations of the well-studied extremely red objects (EROs) HR10 and LBDS53W091 from 3.6 to 160 microns. These galaxies are the prototypes of the two primary classes of EROs: dusty starbursts and old, evolved galaxies, respectively. Both galaxies, as well as LBDS53W069, another example of an old, quiescent galaxy, are well-detected out to 8 microns. However, only the dusty starburst HR10 is detected in the far-infrared. All three EROs have stellar masses of a few times 10^11 M(sun). Using evolutionary model fits to their multiband photometry, we predict the infrared colors of similar EROs at 1<z<2. We find that blueward of observed 10 microns, the two ERO classes are virtually indistinguishable photometrically. Deep spectroscopy and 24 micron data allow the classes to be separated.