Good question. We do have the FlangeFeature.BendFeature property, which will give is a single BendFeature (not to be confused with the simpler Bend object). The BendFeature object does not have any direct properties about any corners. Then, if we dig down ingo the BendDefinition of that BendFeature, we still don't see any direct properties about corners. Then if we dig even deeper into the BendDefinition's BendOptions (through the BendDefinition.BendOptions property), we see some settings about 'bend transition, bend relief shape, bend relief size, and such, but nothing about corners.
Now back to the FlangeFeature again, it does not have a direct property for a CornerFeature, like it had for its BendFeature. And it does not seem to have any other direct properties which have anything to do with corners. Then when we step down into the FlangeDefinition, we have the FlangeDefinition.CornerOptions property, which gives us a CornerOptions object, which is where the can set the 'default' specifications for corner relief location, shape, & size. But that is just a single set of 'default' settings that apply to 'all' the corners that the feature creates, and does not offer us a way to specify unique 'overrides' for each individual corner.
When we select that browser node under the FlangeFeature in the model browser tree labeled 'Corner', it highlights all 4 corners which that feature created, not just one of them. And there are not 4 of those nodes, which would mean one for each corner. So, it apparently does not represent a single CornerFeature, or multiple CornerFeatures, according to the API, because when I inspect the total number of CornerFeatures in the SheetMetalFeatures.CornerFeatures.Count, there are zero, not even one, after that FlangeFeature has been created, and the document updated. It seems to represent a entry way to allow us to either apply individual edits/overrides to each corner directly, or reset all of those individual corner edits/overrides. Generally speaking, pretty much every user interface tool will have a ControlDefinition, also known as 'commands' in some scenarios. This is where its name is defined, the icons involved, and has the Execute method for running a block of code when it gets interacted with. In this case, there are 3 such ControlDefinitions, whose names I will list below:
SheetMetalFlangeCornerResetCtxCmd
SheetMetalFlangeEditCtxCmd
SheetMetalFlangeCornerEditCtxCmd
The middle one there is the first that happens when we choose that 'Edit Corners' option in the context menu. Then the last one in the list is the second to happen, just after that, but pretty much at the same time. Then when we choose the 'Reset All Corners' option, it executes that first command listed there.

Wesley Crihfield

(Not an Autodesk Employee)