The unprecedented imaging power of James Webb Space Telescope (JWST) provides new abilities to observe the shapes of objects in the early Universe in a way that has not been possible before. Recently, JWST acquired a deep field image inside the same field imaged in the past as the Hubble Space Telescope (HST) Ultra Deep Field. Computer-based quantitative analysis of spiral galaxies in that field shows that among 34 galaxies for which their rotation of direction can be determined by the shapes of the arms, 24 rotate clockwise, and just 10 rotate counterclockwise. The one-tailed binomial distribution probability to have asymmetry equal or stronger than the observed asymmetry by chance is $\sim$0.012. While the analysis is limited by the small size of the data, the observed asymmetry is aligned with all relevant previous large-scale analyses from all premier digital sky surveys, all show a higher number of galaxies rotating clockwise in that part of the sky, and the magnitude of the asymmetry increases as the redshift gets higher. This paper also provides data and analysis to reproduce previous experiments suggesting that the distribution of galaxy rotation in the Universe is random, to show that the exact same data used in these studies in fact show non-random distribution, and in excellent agreement with the results shown here. These findings reinforce consideration of the possibility that the directions of rotation of spiral galaxies as observed from Earth are not necessarily randomly distributed. The explanation can be related to the large-scale structure of the Universe, but can also be related to a possible anomaly in the physics of galaxy rotation.