The Basque people of northern Spain loom large in any attempt to understand the ethnogenesis of European populations. That is because the speak the only indubitably indigenous non-Indo-European language in Western Europe. By this, I mean that other potential non-Indo-European languages such as the Tartessian or Pictish are either doubtful or uncertain in classification and understanding (Etruscan, or later Arabic in Spain, are not indigenous). As such they may lay claim to be representatives of a deeper pre-Indo-European substratum in the European population. Up until a few years ago the conventional view was that the Basques were the scions of the Paleolithic peoples of Western Europe, who eventually took up farming. Whether by chance or necessity, they alone retained the language of the original Europeans.
That view is now being challenged. It may be that the Basques were the descendants of the first emigres from the Near East, sweeping along the Mediterranean and Atlantic fringe, searching for arable land. This is no new orthodoxy, but if you grant this, then it opens up the likelihood that the European palimpsest is very complicated indeed. Of the public genome blogging projects Dienekes Pontikos has the most extensive database and greatest geographic coverage in the Dodecad Ancestry Project. He has repeatedly found that the Basques, along with the Finns, seem to lack a genetic element present in low amounts in other European populations, more or less, with a modal value in the trans-Caucasian region. The Spanish and the French as a whole exhibit admixture with this element, only their Basque minorities seem to lack it. The inference one can make from this is clear: these indigenous non-Indo-European peoples lack this component because it was introduced by Indo-Europeans! Now Dienekes has looked at the IBS (Spanish) sample from the 1000 Genomes, and the Basque distinctiveness jumps out again:
The labels are Dienekes', so don't take them literally. Additionally, remember that each component is a stylized representation in many cases, not a real ancestral component (e.g., one component might really be a hybrid). Rather, the slices give us a sense of population relationships. I claim from these results that the Basques are distinctive, but not distinct. The clearly overlap with other Spanish populations, more or less, but they stand out in having a far lower proportion of the "Caucasus" element. However you label this component, you generally see a similar deficiency in the Finns. Again, the Finns and the Swedes overlap, but the Swedes are distinctive in having this element at low proportions. Please note that quite often this component also differentiates Indo-European speaking South Asians from non-Indo-European ones! I think a "virtual genomes" method would be very interesting in exploring the commonalities across these populations.
Secondarily, also note that the Basque seem to lack the Northwest African/Southwest Asian component common in the other groups. This element exists even among other northern Spanish groups. The Basque exemption from tests of limpeza de sangue does seem justified; the only inference I can reasonably make is that the Moorish occupation left a substantial genetic impact on the peninsula. Far more than I would have thought. The standard 'culturalist' model is that the Spanish Muslims were predominantly converts, and their conversion back to Christianity between 1100 and 1600 (as Islam receded in Iberia) was a reversion. But these results lean me toward the proposition that many people with substantial Arab and Berber ancestry became Spanish Christians in the years of the reconquest. Such a substantial demographic impact due to what was a military conquest of a relatively small number (at least initially) increases my own suspicion that the Basques themselves are not the descendants of Paleolithic Iberians in totality.