When Pradosham falls on a Saturday, it’s a qualitatively different event.
When Trayodashi coincides with Saturday, the observance is called Shani Pradosham. Shani is Saturn of course, and from him the name Shani-varam or Saturday comes. Shani (சனி) is not simply a malefic deity who causes misfortune, but he is a dispenser of justice who compels people to confront the consequences of their karma. Since Pradosham itself is a period when one can worship Siva to wipe away the doshas or faults of past actions, the coming together of Pradosha with Shani-kizhamai is considered doubly auspicious or potent. Incidentally, this is not just a coincidence of Pradosham and Saturday, it is the specific occurrence of trayodashi during the twilight hour, based on tithi and sunset calculations.
Here’s the process: 330pm abhishekam for Vrushabha and Siva (and Ambal, usually), 6pm ganapati puja, 630pm ul-purappadu with Pradoshanayakar, who circles the main temple three times in its prakaram, first to Vadyam, then to Vedam, and finally to Tamil recitations of Sivapuranam, Tevaram etc. Those last are always a chorus of devotees' voices. At Nandhikeswara the process is much the same, except that Pradosham is always accompanied by Kailaya Vadyam–and that makes for a very different atmosphere. Plus of course, the readings here on second and third rounds are all in Tamil, women are given a chance to carry the deity and there’s a fair bit more dancing!