We introduce a new entanglement measure for two-qubit states that we call the
binegativity. This new measure is compared to the concurrence and negativity,
the only other known entanglement measures that can be computed analytically
for all two-qubit states. The ordering of entangled two-qubit mixed states
induced by the binegativity is distinct from the ordering determined by
previously known measures, yielding new insights into the structure of
entangled mixed states of two qubits. In addition, we also provide a simple,
alternate proof of a known fact related to the binegativity of two qubits.