A particularly simple Lorentz-violating modification of the Maxwell theory of photons maintains gauge invariance, CPT, and renormalization. This modified-Maxwell theory, coupled to standard Dirac particles, involves nineteen dimensionless "deformation parameters." Ten of these parameters lead to birefringence and are already tightly constrained by astrophysics. New bounds on the remaining nine nonbirefringent parameters can be obtained from the absence of vacuum Cherenkov radiation for

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... -energy cosmic rays (UHECRs). Using selected UHECR events recorded at the Pierre Auger Observatory and assigning pseudo-random directions (i.e., assuming large-scale isotropy), Cherenkov bounds are found at the 10^-18 level, which improve considerably upon current laboratory bounds. Future UHECR observations may reduce these Cherenkov bounds to the 10^-23 level. An Addendum with two-sided bounds has been published separately [Phys. Rev. D 77, 117901 (2008), arXiv:0806.4351].