Integrated sensing and communication (ISAC) is regarded as a promising technology for next-generation communication networks. As the demand for communication performance significantly increases, extremely large-scale antenna arrays and tremendously high-frequency bands get widely applied in communication systems, leading to the expansion of the near-field region. On a parallel track, movable antennas (MAs) and six-dimensional MAs (6DMAs) are proposed as emerging technologies to improve the performance of communication and sensing. Based on such a background, this paper investigates the performance of ISAC systems in the near-field region, focusing on a novel system architecture that employs rotatable MAs (RMAs). Additionally, a spherical wave near-field channel model with respect to RMAs' rotations and positions is derived by considering the effective aperture loss. Two designs are explored: a sensing-centric design that minimizes the Cram\'er-Rao bound (CRB) with signal-to-interference-plus-noise ratio (SINR) constraints, and a communication-centric design that maximizes the sum-rate with a CRB constraint. To solve the formulated optimization problems, the paper proposes two alternating optimization (AO) based algorithms composed of the semidefinite relaxation (SDR) method and the particle swarm optimization (PSO) method. Numerical results demonstrate the convergence and effectiveness of the proposed algorithms and the superiority of the proposed setups for both sensing and communication performance compared to traditional fixed antenna systems, highlighting the potential of RMAs to enhance ISAC systems in near-field scenarios.