Code LLMs often portray inconsistent program behaviors. Developers typically employ benchmarks to assess Code LLMs, but most benchmarks are hand-crafted, static and do not target consistency property. In this work, we pose the scientific question: how can we automatically discover inconsistent program behaviors in Code LLMs? To address this challenge, we propose an automated consistency testing method, called MUCOCO, which employs semantic-preserving mutation analysis to expose inconsistent behaviors in code LLMs. Given a coding query, MUCOCO automatically transforms its program into semantically equivalent programs (aka mutants) and detects inconsistencies between the mutants and the original program (e.g., different output or test failure). We evaluate MUCOCO using four (4) coding tasks and seven (7) LLMs. Results show that MUCOCO is effective in exposing inconsistency and outperforms the closest baseline (TURBULENCE). About one in seven (15%) inputs generated by MUCOCO exposed inconsistencies. Our work motivates the need to test Code LLMs for consistency property