In this work, we propose a Unified framework of Sequential Search and Recommendation (UnifiedSSR) for joint learning of user behavior history in both search and recommendation scenarios. Specifically, we consider user-interacted products in the recommendation scenario, user-interacted products and user-issued queries in the search scenario as three distinct types of user behaviors. We propose a dual-branch network to encode the pair of interacted product history and issued query history in the search scenario in parallel. This allows for cross-scenario modeling by deactivating the query branch for the recommendation scenario. Through the parameter sharing between dual branches, as well as between product branches in two scenarios, we incorporate cross-view and cross-scenario associations of user behaviors, providing a comprehensive understanding of user behavior patterns. To further enhance user behavior modeling by capturing the underlying dynamic intent, an Intent-oriented Session Modeling module is designed for inferring intent-oriented semantic sessions from the contextual information in behavior sequences. In particular, we consider self-supervised learning signals from two perspectives for intent-oriented semantic session locating, which encourage session discrimination within each behavior sequence and session alignment between dual behavior sequences. Extensive experiments on three public datasets demonstrate that UnifiedSSR consistently outperforms state-of-the-art methods for both search and recommendation.