We present analysis and modeling of X-ray spectra from the blast wave shock of DEM L71 in the Large Magellanic Cloud. This remnant exhibits widespread Balmer-dominated emission characteristic of nonradiative shocks in partially neutral gas. We have used Chandra ACIS-S data and optical Fabry-Perot spectra of the blast wave to measure the electron and proton temperatures, respectively. In principle, when combined, these measurements can determine the degree of electron-ion temperature equilibration (g(0) equivalent to T-e/T-p) immediately behind the shock front. In our X-ray analysis we fit Chandra spectra of three nested regions behind the blast wave under three different scenarios: (1) a planar, initially unequilibrated shock (g(0) = m(e)/m(p)), where the downstream electron and proton temperatures equilibrate through Coulomb collisions, (2) a planar, immediately equilibrated shock (g(0) = 1), and (3) a spherical, equilibrated shock under Sedov evolution. Using independent measurements of T-e and T-p, we find that the X-ray spectra from the fastest blast wave locations (V-s similar to 700-1000 km s(-1)) are consistent with little or no equilibration at the shock front and are inconsistent with full equilibration. In contrast, spectra from regions showing slower blast wave speeds (V-s similar to 400-600 km s(-1)) allow full equilibration but exclude zero equilibration. In order to directly constrain the initial equilibration, we incorporated our knowledge of the proton temperatures into our X-ray models to build planar shock models that allow for a variable g(0). This model confirmed and strengthened the above results. Specifically, we found that X-ray spectra from an intermediate-velocity shock (V-s similar to 800 km s(-1)) were consistent with intermediate equilibration, excluding both g(0) = m(e)/m(p) and g(0) = 1 at greater than 1 sigma. Overall, our results support the picture of decreasing electron-ion equilibration with increasing shock speed found from previous studies of optical spectra in other Balmer-dominated supernova remnants.