Long-slit stellar kinematic observations were obtained along the major axis of 30 edge-on spiral galaxies, 24 with a boxy or peanut-shaped (B/PS) bulge and six with other bulge types for comparison. Such B/ PS bulges are identified in at least 45% of highly inclined systems, and a growing body of theoretical and observational work suggests that they are the edge-on projection of thickened bars. Profiles of the mean stellar velocity V, the velocity dispersion , as well as the asymmetric (h 3 ) and symmetric (h 4 ) deviations from a pure Gaussian are presented for all objects. Comparing these profiles with stellar kinematic bar diagnostics developed from N-body simulations, we find bar signatures in 24 of our sample galaxies (80%). Galaxies with a B/ PS bulge typically show a double-humped rotation curve with an intermediate dip or plateau. They also frequently show a rather flat central velocity dispersion profile accompanied by a secondary peak or plateau, and numerous galaxies have a local central minimum (k40%). The h 3 profiles display up to three slope reversals. Most importantly, h 3 is normally correlated with V over the presumed bar length, contrary to expectations from axisymmetric disks. These characteristic bar signatures strengthen the case for a close relationship between B/ PS bulges and bars and leave little room for other explanations of the bulges' shape. We also find that h 3 is anticorrelated with V in the very center of most galaxies (k60%), indicating that these objects additionally harbor cold and dense decoupled (quasi-) axisymmetric central stellar disks, which may be related to the central light peaks. These central disks coincide with previously identified star-forming ionized-gas disks (nuclear spirals) in gas-rich systems, and we argue that they formed out of gas accumulated by the bar at its center through inflow. As suggested by N-body models, the asymmetry of the velocity profile (h 3 ) appears to be a reliable tracer of asymmetries in disks, allowing us to discriminate between axisymmetric and barred disks seen in projection. B/ PS bulges (and thus a large fraction of all bulges) appear to be made up mostly of disk material, which has acquired a large vertical extent through bar-driven vertical instabilities. Their formation is thus probably dominated by secular evolution processes rather than merging.