We present images of the outflow from Cep E in the 1-0 S(1), the 2-1 S(1), and, for the first time in any outflow, the 3-2S(3) line of molecular hydrogen. These are supplemented by [Fe II]1.64 mu m and narrow-band continuum images. We find two almost perpendicular outflows emanating from Cep E. From these and from the spectral energy distribution the Cep E source appears to be a Class 0 binary. Wiggles and sideways positional offsets of bows in the flow are interpreted as due to precession. A crude precession model then yields estimates for the precession angle, precession period, and possibly also for the binary separation. Line ratio maps, built from the molecular hydrogen images, yield the excitational state of the H-2 gas. Surprisingly, the 2-1 S(1)/1-0 S(1) and the 3-2 S(3)/1-0 S(1) line ratios are largely constant everywhere in the outflow. Models of slabs of gas at one single or two constant temperatures fail to account for our data, as do fluorescence, and planar J- and C-shock models. Moreover, J-type bow shocks are a possible, though not very probable, explanation for our data, since they work only if the bows are extremely wide. C-type bow shocks, on the other hand, explain our data well. (C) 1996 American Astronomical Society.