Viscous hydrodynamic calculations of high energy heavy-ion collisions (Nb-Nb and Au-Au) from 200 to 800 MeV/nucleon are presented. The resulting baryon rapidity distributions, the in-plane transverse momentum transfer (bounce-off), and the azimuthal dependence of the midrapidity particles (off-plane squeeze out) compare well with Plastic Ball data. We find that the considered observables are sensitive both to the nuclear equation of state and to the nuclear shear viscosity [eta]. Transverse momentum distributions indicate a high shear viscosity ([eta][approx]60 MeV/fm[sup 2] [ital c]) in the compression zone, in agreement with nuclear matter estimates. The bulk viscosity [zeta] influences only the entropy production during the expansion stage; collective observables like flow and [ital dN]/[ital dY] do not depend strongly on [zeta]. The recently observed off-plane ([phi]=90[degree]) squeeze-out, which is found in the [ital triple]-[ital differential] rapidity distribution, exhibits the strongest sensitivity to the nuclear equation of state. It is demonstrated that for very central collisions, [ital b]=1 fm, the squeeze-out is visible even in the double-differential cross section. This is experimentally accessible by studying azimuthally symmetric events, as confirmed recently by data of the European 4[pi] detector collaboration at Gesellchaft fuer Schwerionforschung Darmstadt.