The strengths of the schemes we will present are their high order of accuracy in both space and time combined with their ability to march in time with a time step at the domain of dependence limit independent of the order. Additionally, the methods are globally super-convergent, i.e. the number of degrees of freedom per cell is (m+1)^d but the methods achieve orders of accuracy 2m. We note that the L2 super-convergence holds globally in space and time, unlike most other spatial discretizations, where super-convergence is limited to a few specific points and often rely on the use of negative norms. Our primary interest of these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type. We provide stability and convergence results for one dimensional periodic domains. The analysis of the conservative method is quite different from the analysis of previous dissipative Hermite methods and introduces a, to our knowledge, novel technique for analyzing conservative schemes for wave equations in second order form. This is joint work with Thomas Hagstrom (SMU) and Arturo Vargas (Rice, LLNL).