Structural characterization by neutron diffraction, supported by magnetic, SHG, and mu+SR data, reveals that the n = 2 Ruddlesden-Popper phase La2SrCr2O7 adopts a highly unusual structural configuration in which the cooperative rotations of the CrO6 octahedra are out of phase in all three Cartesian directions (Phi Phi Phi(z)/Phi Phi Phi(z);a(-)a(-)c(-)/a(-)a(-)c(-)) as described in space group A2/a. First-principles DFT calculations indicate that this unusual structural arrangement can be attributed to coupling between the La/Sr A-site distribution and the rotations of the CrO6 units, which combine to relieve the local deformations of the chromium oxygen octahedra. This coupling suggests new chemical "handles" by which the rotational distortions or A-site cation order of Ruddlesden-Popper phases can be directed to optimize physical behavior. Low-temperature neutron diffraction data and mu+SR data indicate La2SrCr2O7 adopts a G-type antiferromagnetically ordered state below T-N similar to 260 K.