LetUbe the set of all binary sequences of periodp=2^{n}-1containing(p+1)/2ones and(p-1)/2zeros per period. There is a lattice of interesting subsets ofU, the smallest of which is the setPN(the maximum-length linear shift register sequences of periodp). In between are sets with the run statistics ofPN, with the correlation properties ofPN, with the "span-nproperty" (that every nonzero subseqnonce of lengthnoccurs in each period), and others. Results concerning the interrelationships of these subsets are obtained, examples are given to show that certain intersections of subsets are nonempty, and conjectures are formulated regarding other intersections of subsets. For example, it is conjectured that all span-nsequences with the two-level autocorrelation property are in classPN. Some relationships between run properties and correlation properties of binary sequences are also obtained.