This connective ending is a combination of the connective ending -고 (and) and the particle 도 (also, even). In this combination, -고 takes the sequential meaning of after or then, and -도 takes the meaning of even.
V-고도 is a contraction of V-고 + 아/어도 , and used to express an that an unexpected action or situation occurs after the completion of the action in the preceding clause (even after, even though)
A-고도 is a contraction of A-V + 아/어도, and used to express that something has two opposite characteristics (and yet, but)
Note that the verb preceding -고도 must be in the present tense, and that the subject of the two clauses must be the same.