All that info is appreciated, Safetygem, but...
I've read ALL of the ANSI helmet standards, and none of them have tests that would be affected in the least if the shell were reversed, or placed in any other orientation with the liner. Every impact test is from directly above the test "head." For anyone -including OSHA- to say otherwise is to adopt the attitude of "everything not specifically permitted is forbidden."

Then there is the matter of "is a hard hat required?" Or, would a 'bump cap' suffice? Or is the helmet simply a part of the uniform?

More important, we must ask if wearing the "hard hat" itself creates a hazard. For example....though there is no such requirement in the ANSI standard....almost all hard hats direct rainfall forward, over the bill. If you were working on an open panel, would you want the hat to direct stuff towards the panel? Are you working in a position where the brim blocks your vision? Does the brim interfere with the use of other safety gear?

What was raised- namely, the wearing of a reversed hard hat as a fashion statement- is properly a management issue, and not a safety one. Do you want to look profesional and competently serious- or do you want folks to treat you like an overaged delinquent?