I suspect you are talking about the Cooper MonoBilt or Swingtop razor. It has a one-piece cap, hinged at the end of the comb guard which opens when you twist the knob on the end of the handle.
It was manufactured by the Cooper Safety Razor Corp. of Brooklyn, NY. It was patented in 1932. The Cooper trademark was first used in 1932, but I haven't been able to establish how long the products were on the market.
There are pictures of some other Cooper models in the L-W price guide on pages 37 and 63, but they don't seem to have a picture of the MonoBuilt. These other Cooper models were three piece razors of the more conventional Gillette design.