In this {it Letter}, we present sensitive millimeter SiO (J=5-4; $ u$=0) line observations of the outflow arising from the enigmatic object Orion Source I made with the Atacama Large Millimeter/Submillimeter Array (ALMA). The observations reveal that at scales of a few thousand AU, the outflow has a marked butterfly morphology along a northeast-southwest axis. However, contrary to what is found in the SiO and H$_2$O maser observations at scales of tens of AU, the blueshifted radial velocities of the moving gas are found to the northwest, while the redshifted velocities are in the southeast. The ALMA observations are complemented with SiO (J=8-7; $ u$=0) maps (with a similar spatial resolution) obtained with the Submillimeter Array (SMA). These observations also show a similar morphology and velocity structure in this outflow. We discuss some possibilities to explain these differences at small and large scales across the flow.