Do English speakers think about time the way they talk about it? In spoken English, time appears to flow along the sagittal axis (front/back): the future is ahead and the past is behind us. Here we show that when asked to gesture about past and future events deliberately, English speakers often use the sagittal axis, as language suggests they should. By contrast, when producing co-speech gestures spontaneously, they use the lateral axis (left/right) overwhelmingly more often, gesturing leftward for earlier times and rightward for later times. This left-right mapping of time is consistent with the flow of time on calendars and graphs in English-speaking cultures, but is completely absent from conventional spoken metaphors. English speakers gesture on the lateral axis even when they are using front/back metaphors in their co-occurring speech. This speech-gesture dissociation is not due to any lack of lexical or constructional resources to spatialize time laterally in language, nor to any lack of physical resources to spatialize time sagittally in gesture. We propose that when speakers are describing sequences of events, they often use neither the Moving Ego nor Moving Time perspectives. Rather, they adopt a “Moving Attention” perspective, which is grounded in patterns of interaction with cultural artifacts, not in patterns of interaction with the natural environment. We suggest possible pragmatic, kinematic, and mnemonic motivations for the use of a lateral mental timeline in gesture and in thought. Gestures reveal an implicit spatial conceptualization of time that cannot be inferred from language.