Trip purpose is essential information supporting many downstream tasks in intelligent transportation systems, such as travel behaviour comprehension, location-based service, and urban planning. The observation of trip purpose is a necessary aspect of travel surveys, as it is difficult to obtain clear annotated purposes by another approach. However, the limitations of sampling volume, survey budget, and survey frequency make it difficult to rely solely on travel surveys in the era of big data. There has long been a demand for methods to accurately infer trip purpose. An accurate, generalizable, and robust inference method for trip purpose can be the solid first step towards wide and diverse applications. Existing studies have made significant efforts to reveal features correlated with the trip purpose and leverage chaining patterns between trips. However, geographic contextual information has not often been considered. The spatial correlations and chaining patterns hidden in travelled zones are worth further exploration. Additionally, complex activity-zone interactions have not been considered in previous models. In terms of the trip chain level, the generation of a trip could not be only correlated to its directly associated zones but also the zones before or after it, and vice versa. Here, we propose a framework-Dual-Flow Attentive Network with Feature Crossing (DACross), specifically for chained trip purpose inference. We form trip chains based on a new modelling perspective that treats trip activities and travelled geographic zones as two chains with interactions. Correspondingly, we propose DACross, which consists of two parallel attentive branches and a co-attentive feature crossing module, for fully learning intra- and inter-chain dependencies. We conducted extensive experiments on four large-scale real-world datasets to evaluate not only the performance of DACross but also the generalizability of the proposed framework among different cities and various scenarios. Experimental results prove the overall superiority of the proposed DACross.