We consider theories consisting of a planar interface with X = 4 super-Yang-Mills on either side and varying gauge coupling across the interface. The interface does not carry any independent degrees of freedom, but is allowed to support local gauge invariant operators, included with independent interface couplings. In general, both conformal symmetry and supersymmetry will be broken, but for special arrangements of the interface couplings, these symmetries may be restored. We provide a systematic classification of all allowed interface supersymmetries. We find new theories preserving eight and four Poincare supersymmetries, which get extended to sixteen and eight supersymmetries in the conformal limit, respectively with SU(2) x SU(2), SO(2) x SU(2) internal symmetry. The Lagrangians for these theories are explicitly constructed. We also recover the theory with two Poincare supersymmetries and SU(3) internal symmetry proposed earlier as a candidate CIFT dual to super-Janus. Since our new interface theories have only operators from the supergravity multiplet turned on, dual supergravity solutions are expected to exist. We speculate on the possible relation between the interface theory with maximal supersymmetry and the near-horizon limit of the D3-D5 system. (c) 2006 Published by Elsevier B.V.