Over 5 yr of hard X-ray (20-60 keV) monitoring of the 7.66 s accretion-powered pulsar 4U 1626-67 with the Compton Gamma Ray Observatory/BATSE large-area detectors has revealed that the neutron star is now steadily spinning down, in marked contrast to the steady spin-up observed during 1977-1989. This is the second accreting pulsar (the other is GX 1+4) that has shown extended, steady intervals of both spin-up and spin-down. Remarkably, the magnitudes of the spin-up and spin-down torques differ by only 15%, with the neutron star spin changing on a timescale |ν/dot ν| ≈ 5000 yr in both states. The current spin-down rate is itself decreasing on a timescale |dot ν/bar ν| ≈ 26 yr. The long-term timing history shows small-amplitude variations on a 4000 day timescale, which are probably due to variations in the mass transfer rate. The pulsed 20-60 keV emission from 4U 1626-67 is well-fitted by a power-law spectrum with photon index γ = 4.9 and a typical pulsed intensity of 1.5 × 10^(-10) ergs cm^(-2) s^(-1). The low count rates with BATSE prohibited us from constraining the reported 42 minute binary orbit, but we can rule out long-period orbits in the range 2 days lesssim Porb lesssim 900 days. We compare the long-term torque behavior of 4U 1626-67 to other disk-fed accreting pulsars and discuss the implications of our results for the various theories of magnetic accretion torques. The abrupt change in the sign of the torque is difficult to reconcile with the extremely smooth spin-down now observed. The strength of the torque noise in 4U 1626-67, ~10^(-22) Hz^2 s^(-2) Hz^(-1), is the smallest ever measured for an accreting X-ray pulsar, and it is comparable to the timing noise seen in young radio pulsars. We close by pointing out that the core temperature and external torque (the two parameters potentially relevant to internal sources of timing noise) of an accreting neutron star are also comparable to those of young radio pulsars.