  We measure the line-of-sight accelerations of 26 binary pulsars due to the
Milky Way's gravitational potential, and produce a 3-dimensional map of the
acceleration field of the Galaxy. Acceleration measurements directly give us
the change in the line-of-sight velocity at present day, without requiring any
assumptions inherent to kinematic modeling. We measure the Oort limit
($\rho_0=0.062\pm0.017$ \msun/pc$^3$) and the dark matter density in the
midplane ($\rho_{0,\textrm{DM}}=-0.010\pm0.018$ \msun/pc$^3$); these values are
similar to, but have smaller uncertainties than previous pulsar timing
measurements of these quantities. Here, we provide for the first time, values
for the Oort constants and the slope of the rotation curve from direct
acceleration measurements. We find that $A=15.4\pm2.6$ km/s/kpc and
$B=-13.1\pm2.6$ km/s/kpc (consistent with results from \textit{Gaia}), and the
slope of the rotation curve near the Sun is $-2\pm5$ km/s/kpc. We show that the
Galactic acceleration field is clearly asymmetric, but due to data limitations
it is not yet clear which physical processes drive this asymmetry. We provide
updated models of the Galactic potential that account for various sources of
disequilibrium; these models are incompatible with commonly used kinematic
potentials. This indicates that use of kinematically derived Galactic
potentials in precision tests (e.g., in tests of general relativity with pulsar
timing) may be subject to larger uncertainties than reported. The acceleration
data indicates that the mass of the Galaxy within the Solar circle is $2.3
\times 10^{11}$ M$_\odot$, roughly twice as large as currently accepted models.
Additionally, the residuals of the acceleration data compared to existing
Galactic models have a dependence on radial position; this trend can be
explained if the Sun has an additional acceleration away from the Galactic
center.
