In this paper, we present an analysis of the emotion-exchange patterns that arise from Twitter messages sent during emergency events. To this end, we performed a systematic structural analysis of the multiplex communication network that we derived from a data-set including more than 1.9 million tweets that have been sent during five recent shootings and terror events. In order to study the local communication structures that emerge as Twitter users directly exchange emotional messages, we propose the concept of emotion-exchange motifs. Our findings suggest that emotion-exchange motifs which contain reciprocal edges (indicating online conversations) only emerge when users exchange messages that convey anger or fear, either in isolation or in any combination with another emotion. In contrast, the expression of sadness, disgust, surprise, as well as any positive emotion are rather characteristic for emotion-exchange motifs representing one-way communication patterns (instead of online conversations). Among other things, we also found that a higher structural similarity exists between pairs of network layers consisting of one high-arousal emotion and one low-arousal emotion, rather than pairs of network layers belonging to the same arousal dimension.