We analyzed two Chandra observations of PSR J2055+2539 for a total integration time of ∼130 ks to measure the proper motion and study the two elongated nebular features of this source. We did not detect the proper motion, setting an upper limit of 240 mas yr⁻¹ (3σ level), which translates into an upper limit on the transverse velocity of ∼700 km s⁻¹, for an assumed distance of 600 pc. A deep Hα observation did not reveal the bow shock associated with a classical pulsar wind nebula, thus precluding an indirect measurement of the proper motion direction. We determined the main axes of the two nebulae, which are separated by an angle of 160.°8 ± 0.°7, using a new approach based on the rolling Hough transformation (RHT). We analyzed the shape of the first 8′ (out of the 12′ seen by XMM-Newton) of the brighter, extremely collimated nebula. Based on a combination of our results from a standard analysis and a nebular modeling obtained from the RHT, we find that the brightest nebula is curved on an arcmin scale and has a thickness ranging from ∼9″ to ∼31″ and a possible (single or multiple) helicoidal pattern. We could not constrain the shape of the fainter nebula. We discuss our results in the context of other known similar features and place particular emphasis on the Lighthouse nebula associated with PSR J1101−6101. We speculate that a peculiar geometry of the powering pulsar may play an important role in the formation of such features.