The isospin-invariant interacting boson model IBM-3 is analyzed in situations where $SU_T(3)$ charge symmetry [or, equivalently, $U_L(6)$ $sd$ symmetry] is conserved. Analytic expressions for energies, electromagnetic transitions, two-nucleon transfer probabilities, and boson-number expectation values are obtained for the three possible dynamical symmetry limits, U(5), SU(3), and O(6). Results found in IBM-3 are related to corresponding ones in IBM-1 and IBM-2. Numerical calculations are presented for $f_{7/2}$-shell nuclei and some features that distinguish IBM-3 from its predecessors IBM-1 and IBM-2 are pointed out.