These are the results of a measurement of inclusive production at high transverse momentum (P perpendicular to) of neutral pions (.pi./sup 0/) from protons by beams of proton (p), kaon (K), pion (.pi.), and antiproton (antiproton) particles. The experiment used the Fermilab M2 beam line at momenta of 100, 200 and 300 GeV/c striking a liquid hydrogen target. The data include center of mass (cm) production angles of the .pi./sup 0/ between 2/sup 0/ and 115/sup 0/ for P perpendicular to greater than 1 GeV/c. A pair of differential Cherenkov counters distinguished the types of beam particles. The photon (.gamma.) detector of a lead and scintillator sandwich measured both photons from the .pi./sup 0/ .-->. 2.gamma. decay and enabled a reconstruction of the .pi./sup 0/ kinematics. The measurements show several remarkable features distinguishing the production rates for these different beams, including a forward peak in the cm for .pi.p collisions and much less .pi./sup 0/ production at high P perpendicular to in pp than in .pi.p or Kp collisions. A simple parametrization accurately represents the data over most of its range, showing approximate scaling in the radial variable x/sub R/ = 2P/.sqrt.s (in the cm). Several parton models are compared with the data. A heuristic picture in terms of the quark constituents of the interacting hadrons accounts for many features of the data in an intuitive way. 62 references.