We report the results of a new 60 ks Chandra X-ray Observatory Advanced CCD Imaging Spectrometer S-array (ACIS-S) observation of the reddened, radio-selected, highly polarized `FeLoBAL quasar FIRST J1556+3517. We investigated a number of models of varied sophistication to fit the 531-photon spectrum. These models ranged from simple power laws to power laws absorbed by hydrogen gas in differing ionization states and degrees of partial covering. Preferred fits indicate that the intrinsic X-ray flux is consistent with that expected for quasars of similarly high luminosity, i.e., an intrinsic, dereddened and unabsorbed optical to X-ray spectral index of -1.7. We cannot tightly constrain the intrinsic X-ray power-law slope, but find indications that it is flat (photon index Gamma = 1.7 or flatter at a >99% confidence for a neutral hydrogen absorber model). Absorption is present, with a column density a few times 10^23 cm^-2, with both partially ionized models and partially covering neutral hydrogen models providing good fits. We present several lines of argument that suggest the fraction of X-ray emissions associated with the radio jet is not large. We combine our Chandra data with observations from the literature to construct the spectral energy distribution of FIRST J1556+3517 from radio to X-ray energies. We make corrections for Doppler beaming for the pole-on radio jet, optical dust reddening, and X-ray absorption, in order to recover a probable intrinsic spectrum. The quasar FIRST J1556+3517 seems to be an intrinsically normal radio-quiet quasar with a reddened optical/UV spectrum, a Doppler-boosted but intrinsically weak radio jet, and an X-ray absorber not dissimilar from that of other broad absorption line quasars.
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