The pulsar PSR J1756−2251 resides in a relativistic double neutron star binary system with a 7.67-h orbit. We have conducted long-term precision timing on more than 9 yr of data acquired from five telescopes, measuring five post-Keplerian parameters. This has led to several independent tests of general relativity (GR), the most constraining of which shows agreement with the prediction of GR at the 4 per cent level. Our measurement of the orbital decay rate disagrees with that predicted by GR, likely due to systematic observational biases. We have derived the pulsar distance from parallax and orbital decay measurements to be $0.73_{-0.24}^{+0.60}\,$ kpc (68 per cent) and <1.2 kpc (95 per cent upper limit), respectively; these are significantly discrepant from the distance estimated using Galactic electron density models. We have found the pulsar mass to be 1.341 ± 0.007 M⊙, and a low neutron star (NS) companion mass of 1.230 ± 0.007 M⊙. We also determined an upper limit to the spin–orbit misalignment angle of 34° (95 per cent) based on a system geometry fit to long-term profile width measurements. These and other observed properties have led us to hypothesize an evolution involving a low mass–loss, symmetric supernova progenitor to the second-formed NS companion, as is thought to be the case for the double pulsar system PSR J0737−3039A/B. This would make PSR J1756−2251 the second compact binary system providing concrete evidence for this type of NS formation channel.