Using a general definition for reciprocity and anti-reciprocity, the validity conditions of popular definitions and results have been examined and clarified for lumped, linear, time-invariant n-ports. With this definition based purely on input-output characteristics, some new criteria are obtained with no additional assumptions, such as the n-port being initially relaxed or possessing a parameter matrix or a state description. This definition classifies all n-ports that previous definitions left unclassified. A related anti-reciprocity concept has also been considered in the same context. It has been shown, in addition, that the world of anti-reciprocal elements is limited to only gyrators, with the transformer being the sole reciprocal and anti-reciprocal multi-port.