As far as I'm aware, after your honor of whatever type is appropriate reaches a certain point you get a nifty badge on your portrait that shows OTHER PLAYERS your status. Other than the initial 'Congratulations!' message you get upon login, is there any way to know if your honor has reached a point of display other than asking other players?
From my observations, honorable opponent is only visible for the opposing team while the other 3 honors are only visible to your team, including yourself.
Teammate, Friendly and Helpful should be visible for you. If it is not, you don't have it.
UPDATE: The recent Elise Patch seems to have changed it so honorable opponent is also visible for teammates, which most likely means it will also be visible to the player himself.
Currently, there is no way to view Honorable Opponent yourself, apart from the initial notification that you have it.
However, it will shortly become possible to view that if it is the only honor you have as it will be displayed to team mates. However this is the lowest priority, so if you have teamwork as well then you get the green ribbon instead (overriding purple) and so on.
From the Riot posts (on twitter and reddit), they are still changing around the implementations of displaying honor and may look at further changes if needed.