  Electron spin resonance data of Cu^{2+} ions in La_{14-x}Ca_{x}Cu_{24}O_{41}
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 ~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 degree 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.
