Byrne's symbols are variations of Oughtred's, contamporaneous with Harriot's, see Cajori, History Of Mathematical Notations, vol. I, p.192. They were adopted by Barrow, Newton's teacher, in his edition of Euclid, ibid., p. 210. Why Byrne went with Barrow's notation c. 1660 is hard to say, by his time Harriot's symbols were canonized by Euler, but their origin was not exactly intuition. Perhaps, he felt that it would be "easier to grasp" by those unburdened by entrenched standard. For example, Lambert in 1782 used < and > "in reverse" in his logic, as inspired by implication, see Lewis, Survey of symbolic logic.
On early experiments with various inequality symbols see Tanner, On the Role of Equality and Inequality in the History of Mathematics, p. 167-8. Incidentally, he attaches a confabulation on why Harriot's symbols look the way they do, which, spelled out or not, probably helped their adoption as "intuitive".
Here is a later example of how our related "intuitive" symbols in set theory could have been reversed. Interpreting inference as inclusion of classes was common in the 19th century. Whatever can be said for the implication pointing in the direction it does, if one chooses that as "intuitive" then their inequality symbols should go the opposite way from our today's standard too. Indeed, if something is in A then it is in B (A ⊃ B) means that A must be smaller than B (A ⊂ B), see Why is there this strange contradiction between the language of logic and that of set theory?
Harriot, or rather his editors, derived < and > from a tattoo on a native American's hand, which was symmetric, so what they chose to stand for what was rather random. Derived from this in Schröder's Vorlesungen set inclusion symbol ⊂ then conflicted with the implication symbol ⊃, as used by Peano and later many others. Gergonne earlier derived a similar symbol from the looks of letter C (in "contained"), i.e. equally randomly, see Earliest Uses of Symbols of Set Theory and Logic.
Peano, for his part, was consistent, and used ⊃ for inclusion, as well as implication, see Tou, Math Origins: The Logical Symbols, so his smaller would be >. Peirce, Schröder's inspiration for much of his content, was consistent the other way, his implication looked something like this —<, see Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table.
Hilbert's later symbol → for implication similarly conflicts with <, especially when the arrowhead is drawn bigger.