The molecular structure of the DNA double helix has been known for 60 years, but we remain surprisingly ignorant of the balance of forces that determine its mechanical properties. The DNA double helix is among the stiffest of all biopolymers, but neither theory nor experiment has provided a coherent understanding of the relative roles of attractive base stacking forces and repulsive electrostatic forces creating this stiffness. To gain insight, we have created a family of double-helical DNA-like polymers where one of the four normal bases is replaced with various cationic, anionic or neutral analogs. We apply DNA ligase-catalyzed cyclization kinetics experiments to measure the bending and twisting flexibilities of these polymers under low salt conditions. Interestingly, we show that these modifications alter DNA bending stiffness by only 20%, but have much stronger (5-fold) effects on twist flexibility. We suggest that rather than modifying DNA stiffness through a mechanism easily interpretable as electrostatic, the more dominant effect of neutral and charged base modifications is their ability to drive transitions to helical conformations different from canonical B-form DNA.