  Adding a second Kondo channel to heavy fermion materials reveals new exotic
symmetry breaking phases associated with the development of Kondo coherence. In
this paper, we review two such phases, the "hastatic order" associated with
non-Kramers doublet ground states, where the two-channel nature of the Kondo
coupling is guaranteed by virtual valence fluctuations to an excited Kramers
doublet, and "composite pair superconductivity," where the two channels differ
by charge 2e and can be thought of as virtual valence fluctuations to a
pseudo-isospin doublet. The similarities and differences between these two
orders will be discussed, along with possible realizations in actinide and rare
earth materials like URu2Si2 and NpPd5Al2.
