XMM-Newton and Chandra observations of the transient 7.8 s pulsar XTE J1829-098 are used to characterize its pulse shape and spectrum, and to facilitate a search for an optical or infrared counterpart. In outburst, the absorbed, hard X-ray spectrum with Gamma = 0.76+/-0.13 and N_H = (6.0+/-0.6) x 10^{22} cm^{-2} is typical of X-ray binary pulsars. The precise Chandra localization in a faint state leads to the identification of a probable infrared counterpart at R.A. = 18h29m43.98s, decl. = -09o51'23.0" (J2000.0) with magnitudes K=12.7, H=13.9, I>21.9, and R>23.2. If this is a highly reddened O or B star, we estimate a distance of 10 kpc, at which the maximum observed X-ray luminosity is 2x10^{36} ergs s^{-1}, typical of Be X-ray transients or wind-fed systems. The minimum observed luminosity is 3x10^{32}(d/10 kpc)^2 ergs s^{-1}. We cannot rule out the possibility that the companion is a red giant. The two known X-ray outbursts of XTE J1829-098 are separated by ~1.3 yr, which may be the orbital period or a multiple of it, with the neutron star in an eccentric orbit. We also studied a late M-giant long-period variable that we found only 9" from the X-ray position. It has a pulsation period of ~1.5 yr, but is not the companion of the X-ray source. Comment: 6 pages, 7 figures. To appear in The Astrophysical Journal