Antiferromagnetic magnons possess a distinctive feature absent in their ferromagnetic counterparts: the presence of two distinct handedness modes, the right-handed (RH) and left-handed (LH) precession modes. The magnon handedness determines the sign of spin polarization carried by the propagating magnon, which is indispensable for harnessing the diverse functionalities. However, the control of coherently propagating magnon handedness in antiferromagnets has remained elusive so far. Here we demonstrate the manipulation and electrical readout of propagating magnon handedness in perpendicularly magnetized synthetic antiferromagnets (SAF). We find that the antiferromagnetic magnon handedness can be directly identified by measuring the inverse spin Hall effect (ISHE) voltage, which arises from the spin pumping effect caused by the propagating antiferromagnetic magnons in the SAF structure. The RH and LH modes of the magnon can be distinguishable particularly when the SAF structure is sandwiched by heavy metals with the same sign of spin Hall angle. Moreover, we succeed in controlling the handedness of propagating antiferromagnetic magnons by tuning the excitation microwave frequency. This work unveils promising avenues for harnessing magnon unique properties in antiferromagnet-based magnonic applications.