Abstract: We analyze several insulators that fall outside the current classification of topological insulators. We show that the topological nature of a state is not based on whether it supports gapless edge modes, and that a new concept, the entanglement spectrum, is required to classify insulators. We show that topological insulators with inversion symmetry can support many of the interesting effects that their time reversal counterparts exhibit: theta vacuum, E dot B effect, monopole image, etc. At the same time they do not exhibit gapless edge modes when an edge/surface is placed on the material. We also provide for a gauge-invariant way to calculate topological invariants which has so far been absent in the literature. Experimental consequences will also be discussed.