By studying reactions involving unequal-mass particles with spin, we show that much of the structure of possible families of conspiring Regge poles follows simply from imposing the t=0 analyticity constraints. These requirements imply the necessity for both daughter and conspirator contributions, where in many cases the daughter poles are themselves conspirators. We discuss the pattern of (t=0) singularity cancellation by daughter trajectories in both the single-parity families and double-parity families for general mass and spin processes and demonstrate that the two have rather different structure. As an application, the reactions ππ-->VV, πN-->VN, and NN-->NN are examined in detail from the point of view of analyticity constraints. We show that in all cases factorization of the (first) daughter residues is consistent with the required analyticity properties of the amplitudes, and that in certain (nonevasive) cases the factorization of the daughter residues is directly implied by factorization for the leading pole when the conspiracy constraints are obeyed. We conclude that our results, based on analyticity (and factorization), complement the group-theoretic O(4) classification; the symmetry and analyticity methods give similar information when they overlap, but supplement each other in certain cases when one method is not readily applicable.