This post is related to others, such as here. Consider an analogue clock:
The movement of the hand clockwise relative to the dial is equivalent to the movement of the dial couter-clockwise relative to the hand. That is, the motion of the hand relative to the dial corresponds to the opposite motion of the dial relative to the hand. This is the reciprocity principle of relative motion.
This means that motion for the space frame and the time frame are opposites. The displacement direction and the dischronment direction move in opposite directions. The velocity of A relative to B for space corresponds to the velocity of B relative to A for time, and vice versa.
The interchange of space and time needs to take this into account. Their vectors are in opposite directions.