Two important subsets of tripartite pure states, namely {\em triseparable} states (i.e. states such that the bipartite density matrices obtained by tracing out any party's subsystem are separable) and {\em triorthogonal} states (i.e. states with a three-party Schmidt-like decomposition) are introduced. It is proved that any triseparable state is triorthogonal and vice-versa. The relevance of this result to quantifying tripartite pure-state entanglement is discussed. It is also proved that any purification of a bipartite bound entangled state with positive partial transpose is {\em tri-inseparable}. This is a new necessary condition for such bound entanglement.