I’ve read about these concepts a few times before, but I don’t understand the importance of them.

How can this be used in a risk management context?

Why is “normal” version of contango / backwardation even important, if you can’t determine / verify it until after some time?

If the speculator’s (who is long forward) has a discount rate = riskless rate (where k = discount rate), then:
F = E(St)*EXP([r-k]T) = E(St)
...i.e., neither normal backwardation nor normal contango

But if asset is positively correlated (i.e., beta > 0, systematic risk) then discount rate exceeds riskless rate and:
F = E(St)*EXP([r-k]T) < E(St)
On this theory (i.e., that long forwards require compensation for the risk premium), the forward ought to be in normal backwardation

Re relevance, well, for starters: normal backwardation/contango speaks to whether the current forward price embeds a risk premium or not. Or specifically, in addition to cost of carry impacts, does the forward price also embed compensation for systematic risk? In this respect it is like the issues around “what determines the forward price?” e.g., convenience yield

David