I have tried many ways to do this with no success, but what I want to do is the following:
GRANT ALTER ON [dbo].[theTable] TO [role]
Bonus if y
GRANT ALTER ON [dbo].[theTable] TO [role]
GO
GRANT ALTER ON object TO principal
is the correct form of the statement in your case.
To view the permissions granted to you on the object, use the fn_my_permissions function like this:
SELECT *
FROM sys.fn_my_permissions('object', 'OBJECT')
;