A haruspex in ancient Rome was a religious official who interpreted omens by inspecting the entrails of sacrificial animals. The haruspices were part of a group of seers or auguries whose official function was not so much to foretell the future as to work out whether the gods approved of some proposed course of political or military action. Nothing of importance was undertaken until the auguries had been consulted. Many omens were actively watched for, such as the flight of birds, the pecking behaviour of sacred chickens, or the sound of thunder. The Romans borrowed these techniques from their predecessors, the Etruscans.
Edward Gibbon, in The Decline and Fall of the Roman Empire, was disdainful of what he saw as the barbarous rites of the period: “Amidst the sacred but licentious crowd of priests, of inferior ministers, and of female dancers, who were dedicated to the service of the temple, it was the business of the emperor to bring the wood, to blow the fire, to handle the knife, to slaughter the victim, and, thrusting his bloody hands into the bowels of the expiring animal, to draw forth the heart or liver, and to read, with the consummate skill of an haruspex, imaginary signs of future events. The wisest of the Pagans censured this extravagant superstition, which affected to despise the restraints of prudence and decency.”
The second part of the word is clearly from Latin specere, to look at, but the first part is more mysterious; it may be related to Sanskrit hir, an artery. The technique is called haruspicy. Another word for it is extispicy, a word whose the first element we do know the origin of — it’s from Latin exta, entrails.