Can oxygen burn in hydrogen?
Could you be more clear on what 'burning in' something means? The chemical reactions require both reactants available, so they need to mix - in what sense is Hydrogen mixing and reacting with Oxygen any different that Oxygen mixing and reacting with Hydrogen?
In some sense, we do tend to control these reactions via the fuel, not the oxidant (oxygen free environments are difficult to reproduce on Earth's surface). However there may be some examples of controlling the mixture via restricting the oxidant in a fuel-rich environment... even though I can't even get through the sentence without that sounding wasteful.
Nitrous Oxide
Would you consider the example of Nitrous Oxide injection in an internal combustion engine an example of this? Pre-nitrous, there is compressed atmospheric oxygen and hydrocarbons at a set ratio, and to further increase the burn a nitrous system will inject either NO alone (dry system) or NO and more hydrocarbon fuel (wet system) to add more oxidants that would be available through the 21% Oxygen in the atmosphere.

The Chemical: https://en.wikipedia.org/wiki/Nitrous_oxide
Internal Combustion application: https://en.wikipedia.org/wiki/Nitrous

Candles
How about a candle? There's an abundant fuel source (wax), and we frequently control the shape and characteristics of the burn with shaped glass, not to mention put it out via starving it of oxidant.
In either case, if you were to, say take a lump of solid oxygen (nevermind the practical difficulties of dealing with temperatures below 54 Kelvin) and maintained a burn with Hydrogen, what you'd be likely to find would be combustion only happening at the edges, where the oxygen was evaporating and mixing with the reduction agent, similar to lighting a can of gasoline on fire (only the vapors are flammable).