  This is the first of two papers to study a new emergent symmetry which
connects orbitally excited heavy baryons to the ground states in the combined
heavy quark and large $N_c$ limit. The existence of this symmetry is shown in a
model-independent way, and different possible realizations of the symmetry are
discussed. It is also proved that this emergent symmetry commutes with the
large $N_c$ spin-flavor symmetry.
