Police do seem to be somewhat lacking when understanding the law with regards to cyclists...
18 months or so ago, I was cycling round Central London in the dark trying to meet up with some mates and needed to go the wrong way on a one-way road to save a lot of time (and potentially getting lost) so got off and pushed the bike along the pavement. I was wearing a hi-viz jacket and gloves, with a 10W fixed-beam front light, additional random-flashing front light, helmet-mounted flashing light, a bright, fixed-beam, red light on my seatpost, a flashing one on my chainstay and another flashing on my rucksack, with reflective detail on my shoe-heels, rucksack and back of my helmet. 
One male (fat, smug, young) and one female (young, attractive, obviously new) officer, who may have been Specials (apt name, that!) were walking the other way when the male officer called to me and told me that I was not allowed to be on the pavement. I politely (but probably with some tone of puzzlement) pointed out that I was not riding but walking but he seemed too stupid to know the difference and said that the only reason he was 'letting me off' was the fact that I was so safely visible...
...all done with a cocky, 'look how tough I am' attitude as though he was trying to show off to the girl. I tried to explain but quickly gave up when I realised what a prick he was. Not worth the effort and I had wasted enough time as it was. 
I have a sister, best mate and bro-in-law who are police officers and they all thought his behaviour was totally out of order....
