In this paper, we study the algebraic structure of additive cyclic codes over the alphabet Fr 2 x Fs 4 = Fr 2Fs 4, where r and s are non-negative integers, F2 = GF(2) and F4 = GF(4) are the finite fields of 2 and 4 elements, respectively. We determine generator polynomials for F2F4-additive cyclic codes. We also introduce a linear map W that depends on the trace map T to relate these codes to binary linear codes over F2. Further, we investigate the duals of F2F4-additive cyclic codes. We show that the dual of any F2F4-additive cyclic code is another F2F4-additive cyclic code. Using the mapping W, we provide examples of F2F4-additive cyclic codes whose binary images have optimal parameters. We also consider additive cyclic codes over F4 and give some examples of optimal parameter quantum codes over F4.