This paper studies codes C over GF(4) which have even weights and have the same weight distribution as the dual code C⊥. Some of the results are as follows. All such codes satisfy C⊥ = C (If C⊥= C, T has a binary basis.) The number of such C's is determined, and those of length ≤14 are completely classified. The weight enumerator of C is characterized and an upper bound obtained on the minimum distance. Necessary and sufficient conditions are given for C to be extended cyclic. Two new 5-designs are constructed. A generator matrix for C can be taken to have the form [I | B], where B⊥ = B. We enumerate and classify all circulant matrices B with this property. A number of open problems are listed.