Electron spin resonance data of Cu2+ ions in La(14)-xCaxCu24O41 single crystals (x = 9,11,12) reveal a very large width of the resonance line in the paramagnetic state. This signals an unusually strong anisotropy of approximately 10% of the isotropic Heisenberg superexchange in the Cu-O chains of this compound. The strong anisotropy can be explained by the specific geometry of two symmetrical 90 degrees Cu-O-Cu bonds, which boosts the importance of orbital degrees of freedom. Our data show the apparent limitations of the applicability of an isotropic Heisenberg model to the low-dimensional cuprates.