As with many things in Tolkien, the exact number depends on which drafts you consider to be canon; there's somewhere between zero and four pairs. This answer is mostly in supplement to Wad Cheber's, but with canon sources added. However, the case of Eluréd and Elurín is more nuanced than Ask Middle-earth makes it seem.
But, easy ones first:
Elladan and Elrohir
The canon is covered in my answer to a similar question.
Amrod and Amras
The seven sons of Fëanor were Maidros [> Maedhros] the tall; Maglor a musician and mighty singer, whose voice was heard far over land and sea; Celegorn [> Celegorm] the fair, and Cranthir [> Caranthir} the dark; and Curufin the crafty, who inherited most of his father's skill of hand; and the youngest Damrod and Diriel [> Amrod and Amras], who were twin brothers alike in mood and face.
History of Middle-earth X Morgoth's Ring Part 3: "The Later Quenta Silmarillion" Chapter 1 "The First Phase" V "Of Eldanor and the Princes of the Eldalie"
And in the published Silmarillion (emphasis mine):
The seven sons of Fëanor were Maedhros the tall; Maglor the mighty singer, whose voice was heard far over land and sea; Celegorm the fair, and Caranthir the dark; Curufin the crafty, who inherited most his father's skill of hand; and the youngest Amrod and Amras, who were twin brothers, alike in mood and face. In later days they were great hunters in the woods of Middle-earth; and a hunter also was Celegorm, who in Valinor was a friend of Oromë, and often followed the Vala's horn.
The Silmarillion III Quenta Silmarillion Chapter 5: "Of Eldamar and the Princes of the Eldalië"
Elrond and Elros
Elrond and Elros seem to have been twins for their entire textual history; the last mention of this is in a note added to The Tale of Years:
528 Elros and Elrond twin sons of Eärendil born.
History of Middle-earth XI The War of the Jewels Part 3: "The Wanderings of Húrin and other Writings not Forming Part of the *Quenta Silmarillion" Chapter V: "The Tale of Years"
The year was later changed to 532, but otherwise remains unchanged. This isn't entirely surprising, since Elros was a late addition to the narrative.
Eluréd and Elurín
This is something Tolkien changed his mind on, repeatedly.
Initially, the sons of Dior were not twins; in The Genealogies, Tolkien writes:
Elboron son of Dior born 192; Elbereth his brother born 195
History of Middle-earth V The Lost Road and Other Writings Part 3 Appendix: "The Genealogies, The List of Names and the Second 'Silmarillion' Map"
"Elboron" and "Elbereth" would later be replaces with "Elrun" and "Eldun", and eventually "Eluréd" and "Elurín."
Later on, in a plot synopsis written while writing Narn i Chîn Húrin, he changed this:
500 Elrun and Eldun twin sons of Dior are born.
History of Middle-earth XI The War of the Jewels Part 3: "The Wanderings of Húrin and other Writings not Forming Part of the *Quenta Silmarillion" Chapter I: "The Wanderings of Húrin"
However, he changed his mind at least one more time, writing very late in his life:
Dior [Beren and Lúthien's] son, it is said, spoke both tongues: his father's, and his mother's, the Sindarin of Doriath. For he said: 'I am the first of the Peredil (Half-elven), but I am also the heir of King Elwë, the Eluchíl.' He gave to his elder son the name Eluréd, that is said to have the same significance, but ended in the Beorian word reda 'heir'; to his second son he gave the name Elurín.
History of Middle-earth XII The Peoples of Middle-earth Chapter 12: "The Problem of Ros"
The Problem of Half-Elves
The OP specifically requested that half-elves be excluded. By that definition we're forced to exclude all but Amros and Amras, since all of the others have a Man somewhere in their ancestry1.
Although I don't want to speculate on the reasons for the OP's restriction, there's no biological reason to distinguish between them. As Tolkien has noted on at least one occasion (Letter 153) that there's no biological distinction between Elves and Men, or at least that they're similar enough to breed:
Elves and Men are evidently in biological terms one race, or they could not breed and produce fertile offspring – even as a rare event
The Letters of J.R.R. Tolkien 153: To Peter Hastings (Draft). September 1954
I recall reading somewhere, though I can no longer remember the primary source, that the differences between Elves and Men are essentially spiritual, rather than biological (this is certainly the case with Elvish immortality, as Tolkien says later in Letter 153). If that's true, then the choice of a half-elf to cast their lot in with the Eldar removes the distinction, at least for our purposes.
1 Elrond and Elros are great-grandsons of Beren (through Elwing) and grandsons of Tuor, son of Huor, of the House of Hador (through Eärendil). Elladan and Elrohir have human ancestry through Elrond. Eluréd and Elurín, as mentioned in an above quote, are grandsons of Beren.