Cross-domain recommendation (CDR) addresses the data sparsity and cold-start problems in the target domain by leveraging knowledge from data-rich source domains. However, existing CDR methods often rely on domain-specific features or identifiers that lack transferability across different domains, limiting their ability to capture inter-domain semantic patterns. To overcome this, we propose SemaCDR, a semantics-driven framework for cross-domain sequential recommendation that leverages large language models (LLMs) to construct a unified semantic space. SemaCDR creates multiview item features by integrating LLM-generated domain-agnostic semantics with domain-specific content, aligned by contrastive regularization. SemaCDR systematically creates LLM-generated domain-specific and domain-agnostic semantics, and employs adaptive fusion to generate unified preference representations. Furthermore, it aligns cross-domain behavior sequences with an adaptive fusion mechanism to synthesize interaction sequences from source, target, and mixed domains. Extensive experiments on real-world datasets show that SemaCDR consistently outperforms state-of-the-art baselines, demonstrating its effectiveness in capturing coherent intra-domain patterns while facilitating knowledge transfer across domains.