The site with the geneologies is back up. I'm going to quote what it says about the ERE-Habsburg connection.
Philip of Swabia, son of Frederick Barbarossa, who contended with Otto of Brunswick for the Empire, had no sons; but the marriages of his four daughters are among the most interesting in European history. In a reconciliation of their feud, his oldest daughter, Beatrice, married Otto himself. But they had no children. The younger daughters, Kunigunde, Marie, and Elizabeth, married King Wenceslas I of Bohemia, Duke Henry III of Lower Lorraine and Brabant, and King & St. Ferdinand III of Castile and Leon, respectively. All of these marriages produced children with living descendants, especially among the Hapsburgs and the royal families of Spain, as can be traced at the linked genealogies. This is all of particular interest because of Philip's wife, Irene, who was a daughter of the Roman Emperor Isaac II Angelus. Isaac, a disastrous Emperor, himself was a great-grandson of the outstanding Emperor Alexius I Comnenus, the restorer of Romania after the Turkish invasion. This means that a large part of modern European royalty have been descendants of the Comneni. My impression is that Roman (Byzantine) Imperial descent for recent royalty has often been claimed through the Macedonians, but the only genuine line seems to be from Macedonian in-laws. On the other hand, descent from the Comneni appears to be well attested and with multiple lines, all from Irene Angelina.