We employed infrared spectroscopy along with complementary lattice dynamics and spin density calculations to investigate pressure-driven local structure distortions in the copper coordination polymer Cu(pyz)F-2(H2O)(2). Here, pyz is pyrazine. Our study reveals rich and fully reversible local lattice distortions that buckle the pyrazine ring, disrupt the bc-plane O-H center dot center dot center dot F hydrogen-bonding network, and reinforce magnetic property switching. The resiliency of the soft organic ring is a major factor in the stability of this material. Interestingly, the collective character of the lattice vibrations masks direct information on the Cu-N and Cu-O linkages through the series of pressure-induced Jahn-Teller axis switching transitions, although Cu-F bond softening is clearly identified above 3 GPa. These findings illustrate the importance of combined bulk and local probe techniques for microscopic structure determination in complex materials.