Still with the OP, this isn't a matter of what an employer can do, it's what they think they can do. If they are wrong and persist, then the employee will be vindicated and in certain circumstances eg discrimination, they may get a big pay out, but many people would like a hassle-free job rather than eventually having the last laugh when they go to sign on.
Cycle helmets are one of those areas where many people "know" the law, even when they don't. The "Police thugs" thread was started after a rider was physically brought to a stop by a police officer to be advised on helmets and hi-viz.
viewtopic.php?f=7&t=82174Blackbike's post covers much of what I've been trying to say, but I'm not talking about convictions, rather about actions which can be connected with the employer; arriving at work and leaving provides that connection. To get this away from the emotive subject of cycle helmets, imagine a works in a residential area. Around here we have any number of former mills and the like, cheek-by-jowl with terraced housing which once housed the workforce. Imagine a dispute over inconsiderate on-street parking by staff which escalates to the point of attracting the attention of the local rag, pictures in the paper etc. Is the employer entitled to tell the workforce they must be good neighbours, because to do otherwise would reflect badly on the company?