If a double-balanced mixer using four diodes lose two diodes, became a single-balanced mixer.
The advantages of a double-balanced mixer (with four diodes) over a single balanced mixer (with two diodes) are: better linearity, improved suppression of spurious products (all even-order products of the LO and/or the RF are suppressed), and better isolation between all ports.
The disadvantage is, the quad mixers requires a higher level LO drive, and needs two transformer baluns, when a two-diode mixer needs only one balun.