The members of the Prayer Book Society, and adherents of SSPX, pray the same prayers, and are fed by the same scriptural readings, between Trinity and Advent. But not, regrettably, on the same Sundays.
These readings are first found in Roman lectionary books going back to the seventh and eighth centuries. The Prayers - collects, secrets ["over the Offerings"], postcommunions, are found in the ancient sacramentaries containing formulae which the liturgical scholars of the first half of the twentieth century were in many cases able to identify as written by particular Roman Pontiffs - not least S Leo the Great and S Gelasius. But neither readings nor prayers seem to have been selected to 'go with' each other, nor is it easy to detect the grounds upon which they were chosen at all - except in a few cases, such as the Gospels for Trinity III and Trinity V, which probably relate to the nearby feast of SS Peter and Paul. (Indeed, in early days Sundays were often thought of as 'before' or 'after' important feasts, such as that of S Lawrence, rather than as forming part of a relentless series marching from Pentecost all the way to Advent). It was this lack of strong control which gave the post-conciliar revisers their excuse to disregard their inheritance. I suspect that the jury is still out on whether discarding what the Western Church had possessed for 1300 years was a good idea.
The reasons why the rite of S Pius V - the traditional or Tridentine rite - and the Anglican Prayer Books are not quite in sync', is twofold. (1) The first Mass in the series for the 'green' Sundays was used differently in different places. Pius V inherited a tradition which used it on the weekdays after Trinity Sunday. The English medieval usage which the Prayer Book inherited and perpetuated used it on the first Sunday after Trinity and the weekdays which followed that Sunday. Hence, English custom puts all the Masses one week later than the Pian custom: the Mass of Trinity V, for example, is the Mass for Pentecost V (a Sunday after Trinity comes obviously a week later than the Sunday with the same number after Pentecost because Trinity is a week later than Pentecost).
(2) The English medieval missals used, on TrinityIII, an ancient Roman collect (which had its secret and postcommunion attached to it) which had dropped out of the books used by the revisers of Pius V. And on Trinity IV they used an ancient Roman Gospel which did not make it into the Pian Missal.
So on and after TrinityV, the Prayer Book is a week later than the Tridentine Missal as regards most of the Mass, but two weeks behind with regard to the Prayers and Gospels. If you are still reading this, apply for your £1m reward. (There are one or two disruptions as regards the graduals, but, O gawd, let's not go into that.)
So if you want to use a Tridentine Missal but keep in sync' with what your Prayer Book fellow Anglicans are doing, not to mention, of course, the pre-reformation English missals, you have otch about a bit in the book. Things are, incidentally, a bit easier if you say the Sunday Office from a Tridentine Breviary; on and after Trinity V all you need to do is to use the collect, third nocturn readings, and the antiphons for the Benedictus and Magnificat, according to this formula: Trinity x = Pentecost x-1.
Anglican revisers in 1980 abandoned the ancient series of collects but around 2000 adopted a series which included many of the old collects on their old (Prayer Book) Sundays. The post-conciliar Roman Rite retains a fair number of the old collects but higglety pigglety, according to an enumeration 'per annum' rather than 'post Pentecosten', and (at the moment in Anglophone countries) in 'translations' which render them unrecognisable.