We propose a new method of analyzing low-energy hadron-hadron scattering designed to reveal internal quark-gluon eigenstates. Our method is a modification of the Wigner-Eisenbud formalism suited to the case of confining boundary conditions. We have identified a matrix function of the energy (which we call P) whose poles and residues correspond to the masses and channel projections of quark-gluon eigenstates calculated with spherical bag boundary conditions. To illustrate the formalism we apply it to the low-energy S-wave scattering of pseudoscalar mesons. We find from the data clear evidence of internal states corresponding to quark-model predictions of Q2Q¯2 states at 0.69 and 1.04 GeV in the I=0 channel, at 0.96 GeV in the I=12 channel, at 1.19 GeV in the I=32 channel, and at 1.04 GeV in the I=2 channel. We believe the I=32 and 2 internal states to be the low-energy exotics long predicted by quark models.