By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around 2k_{B}T/h≈1 GHz, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a 1/f power law that matches the magnitude of the 1/f noise near 1 Hz. The antisymmetric component displays a 1/T dependence below 100 mK, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells.