The Centro Cervantino is the town's small museum dedicated to the Cervantes birthplace controversy and to the world of Don Quixote more broadly. The case for Alcázar de San Juan rests primarily on a baptismal record in the church of Santa María la Mayor that names a "Miguel, son of Blas de Cervantes Saavedra," dated 9 November 1558. Alcalá de Henares counters with a record of 9 October 1547 — and the majority of scholars accept Alcalá's claim.
The museum presents the documentary evidence on both sides, along with editions of Don Quixote, illustrations from various centuries, and material relating to the life and times of Cervantes. Even if you remain unconvinced by Alcázar's claim, the context it provides — the rural La Mancha economy, the role of the nobility, the literary milieu of 16th-century Spain — is useful before you head out to see the windmills. The exhibits are largely in Spanish.