The construction industry faces productivity stagnation, skilled labor shortages, and safety concerns. While robotic automation offers solutions, construction robots struggle to adapt to unstructured, dynamic sites. Central to this is improvisation, adapting to unexpected situations through creative problem-solving, which remains predominantly human. In construction's unpredictable environments, collaborative human-robot improvisation is essential for workflow continuity. This research develops a six-level taxonomy classifying human-robot collaboration (HRC) based on improvisation capabilities. Through systematic review of 214 articles (2010-2025), we categorize construction robotics across: Manual Work (Level 0), Human-Controlled Execution (Level 1), Adaptive Manipulation (Level 2), Imitation Learning (Level 3), Human-in-Loop BIM Workflow (Level 4), Cloud-Based Knowledge Integration (Level 5), and True Collaborative Improvisation (Level 6). Analysis reveals current research concentrates at lower levels, with critical gaps in experiential learning and limited progression toward collaborative improvisation. A five-dimensional radar framework illustrates progressive evolution of Planning, Cognitive Role, Physical Execution, Learning Capability, and Improvisation, demonstrating how complementary human-robot capabilities create team performance exceeding individual contributions. The research identifies three fundamental barriers: technical limitations in grounding and dialogic reasoning, conceptual gaps between human improvisation and robotics research, and methodological challenges. We recommend future research emphasizing improved human-robot communication via Augmented/Virtual Reality interfaces, large language model integration, and cloud-based knowledge systems to advance toward true collaborative improvisation.