This critical perspective review examines five major classes of carbon-sequestering cementitious construction materials: accelerated carbonation curing systems, geopolymers and alkali-activated materials, recycled cement derived from construction and demolition waste, biochar-modified composites, and reactive magnesia cements. The review draws principally on peer-reviewed literature published between January 2023 and early 2026. A strict distinction is made throughout between emissions avoidance, carbon capture and utilisation, and carbon dioxide removal, since failure to separate these categories continues to inflate climate claims in this field. Reported sequestration values are converted, where possible, to common functional units under explicitly stated assumptions, while the accounting level of each claim — binder, concrete, or composite — is identified in the corresponding tables and figures. Technology readiness is assessed through an evidence matrix covering laboratory validation, pilot deployment, commercial installations, standards availability, and field durability data. An assumptions-driven scenario analysis, using conservative, moderate, and optimistic variants, suggests that these material classes could contribute approximately 0.4–3.5 Gt CO₂/year by mid-century through combined emissions avoidance and carbon removal. However, strict atmospheric carbon dioxide removal is bounded at a substantially lower level, approximately 0.1–1.5 Gt CO₂/year, and remains constrained by feedstock availability, low-carbon CO₂ supply, processing energy, standardisation gaps, and unresolved durability evidence. Durability is treated not as a secondary technical issue but as a central condition for structural adoption. Particular attention is given to reinforcement corrosion risks in carbonated systems, efflorescence and alkali leaching in alkali-activated binders, strength penalties at higher biochar dosages, and moisture sensitivity in MgO-based systems. The regulatory analysis considers the revised EU Energy Performance of Buildings Directive, the EU Taxonomy, and investor disclosure frameworks, while extending the discussion to China, India, ASEAN countries, the Gulf region, and sub-Saharan Africa, where most future cement demand growth is expected to occur. The 2026 energy-price shock is discussed only as an illustrative stress test of a broader structural vulnerability: the high exposure of clinker-based materials to energy-price volatility. Overall, the review argues that carbon-sequestering cementitious materials can make a meaningful contribution to climate mitigation only if sequestration claims are accounting-transparent, durability-verified, and deployment-ready under real construction-market conditions.