A symmetric Talbot-Lau neutron grating interferometer has been developed for achieving high phase sensitivity. The gratings for the interferometer have been designed by maximizing the intergrating distances available at the experimental facility and optimizing the period of the gratings. The phase sensitivity in a Talbot-Lau grating interferometer has been mathematically modeled and analyzed and compared with experimental data. Evaluation experiments have been performed at the cold neutron imaging facility of the NIST Center for Neutron Research. The symmetric Talbot-Lau neutron grating interferometer exhibits distinct advantages in terms of high sensitivity and wide range for dark-field contrast imaging. Also, the fabrication of gratings is made easier, simpler, and more economical by the Gadox powder filling method, which gives them excellent phase contrast compared with other geometric Talbot-Lau neutron grating interferometers.