We study a flux qubit consisting of a symmetrical pair of superconducting loops, with two Josephson junctions in each, joined by a common Josephson junction—a ‘twin’ flux qubit. The qubit is capacitively coupled to a transmission line, which allows us to characterize the spectrum of the device by measuring the scattering of propagated electromagnetic waves. We perform a detailed analytical analysis of the double-loop system, revealing its properties, and compare experimental results with numerical simulations. At half-flux quantum bias of both loops, the qubit is protected against global and local magnetic field fluctuations with much less sensitivity to the global field in the second order. The system selection rules allow even-odd transitions and prohibit transitions between even-even or odd-odd levels due to the symmetry of the device.