The Congress of Vienna was an unprecedented event; representatives of all the European nations meeting together in one place to resolve the (very great, following the French Revolution, the Napoleonic era and the dissolution of the Holy Roman Empire) issues of the day, rather than all being done by correspondence between individual nations. It did resolve those issues, to the satisfaction of most if, inevitably, not all, and managed to establish a European political order stable enough that it would be 99 years before general war broke out again.
The political map of Europe today has been reshaped by subsequent events, but is still underpinned by the dispositions made at the Congress and set out in painstaking detail in its Final Act, signed nine days before Napoléon I’s also final defeat at Waterloo. And of course the model established in Vienna in 1814-15 has been followed many times since; it could well be argued that the United Nations is a spiritual descendant of the Congress, apart from the numerous such meetings, these days usually attended by heads of government rather than plenipotentiaries, which have taken place over the nearly two centuries since the Final Act was signed.
Three charts follow showing the relationships of the sovereigns of the day, the representatives of eight of whom were the Act's signatories. The split is Catholics with Catholics, Protestants and Orthodox with the same, and then the two sets with each other. Two tables of combined statistics and a note on posterities complete the thread.
An interesting feature of the relationships can be seen in the three keys and the tables of combined statistics. There is no overlap whatsoever, each key is unique. Catholics are closely related through other Catholics, Protestants and Orthodox are closely related through other Protestants (in one case through the Catholic FIIW, but see below for him), and the two sets are for the most part distantly related through Protestants. As far as I can see the only Catholic in the key for the third chart is ELF, Egon VIII of Fürstenberg, though I am not quite certain of the religious affiliation of one or two of the more obscure names.
Protestant princesses were it seemed more willing than Catholics to convert, or at least allow their children to be brought up in the other faith. There was also conversion by princes, whether opportunistic, as with more than one Wittelsbach who read a family tree and saw a good chance of the inheritance of Catholic family lands, or sincere, as with Duke Johann Friedrich of Brunswick-Lüneburg, who is responsible for the prominence in the key of his father Georg (GBL). Another opportunistic conversion was that of Augustus the Strong, Elector of Saxony and thus heir of a line of great Protestant champions, who became Catholic in a successful pursuit of the Polish crown. This act ultimately brought much Protestant blood into senior Catholic lines through his son August III of Poland, A3P in the first chart key.
And yet another was that of Karl Alexander, Duke of Württemberg. At a time when he had no immediate prospect of succeeding to the Duchy he became Catholic in order to further his career in the Austrian military, and married appropriately. This did not result in Protestant blood entering Catholic lines, but rather the reverse. His son Friedrich II Eugen (FIIW, mentioned above) was raised Catholic, but married a Protestant and agreed, Protestantism being the majority religion of Wurttemberg, to his wife raising their children in her faith. It was through him that the Catholic Egon VIII had descendants among the Protestant sovereigns of the day.
I will conclude with one more example, this one of a Protestant princess marrying (and indeed becoming) Catholic. Georg II of Hesse-Darmstadt, GIIHD in the third chart key, was very much a Lutheran. He nevertheless allowed his daughter Elisabeth Amalie to marry the Catholic Philipp Wilhelm, Count Palatine of Neuburg (PEP), a widower nearly 20 years her senior who was expected to eventually succeed as Elector Palatine, and did. Despite the disparity in their ages the marriage was extremely happy, and the couple had no fewer than 17 children.
An exceptionally beautiful woman, Elisabeth Amalie passed on her looks to her daughters, which no doubt assisted in the grand marriages made for them, one becoming Empress, two more becoming Queens of Spain and Portugal respectively, and a fourth Duchess of Parma (yet a fifth daughter, Hedwig Elisabeth, married Jakub, Prince Sobieski, and was the grandmother of Charles Edward Stuart and Cardinal York). The one who was Queen of Spain had the misfortune that her husband was Carlos II, but the others had issue and their descent is widespread today, and indeed was even then, another means of tying Protestant and Catholic sovereigns together.
An explanation of how to read the charts can be found here.