Who first suggested that the area enclosed by a closed path and that enclosed by that path traversed in reverse could be regarded as equal in magnitude but opposite in sign?
Cauchy must have noticed this in connection with contour integrals. Still, perhaps Gauss or someone else came up with this perspective first, possibly in connection with surveying (the first mechanical planimeter was invented in 1818).
I would appreciate not just an attribution of priority but a supporting citation.
Signed areas came up in connection with winding numbers even before Gauss, see Sunada, From Euclid to Riemann and Beyond:
"Gauss briefly noticed in the letter to Bessel in 1811 , the winding number reveals the nature of the complex logarithm. Winding numbers appeared in the study of signed areas by Albrecht Ludwig Friedrich Meister (1724-1788)."
Meister's Generalia de genesifigurarum planarum, et inde pendentibus earum affectionibus (Novi Commentarii Soc. Reg. Scient. Gott., 1 (1769/1770), 144-180) had very few equations, but derived what is now called "shoelace formula". Gauss used the idea of winding numbers in his proofs of FTA (1799), see Eisermann, Fundamental Theorem of Algebra Made Effective.
Cauchy came to signed areas, and volumes, around 1812-13 from a different perspective, not in connection with winding numbers and contour integrals. It was the usual byproduct of algebraic treatment of geometric objects - algebra naturally uniformizes geometrically distinct cases in theorems. Chasles inserted signed segments into the cross-ratio for similar reasons a bit later (1830). Here is Boyer, History of Analytic Geometry:
In 1812-1813, for example, J.P.M. Binet (1786-1856) wrote a long and tedious paper in which he made use of analogous notations (which he called "resultants") in analytic theorems on volumes, areas, and lengths of rectangular configurations in three dimensions. His work includes the equivalent of the multiplication of determinants; but in this early use of determinants the familiar square array was missing. This element, together with the double-subscript notation, was supplied by Cauchy at the very time of Binet's paper. Cauchy applied his "resultants" in attributing signed values to angles, areas, and volumes in coordinate geometry."
Adding to Conifold's excellent answer and stretching the scope of the question, the first to fully understand the importance of signed lines, areas, volumes and hypervolumes in any dimension is very likely Hermann Grassmann and his "Die lineale Ausdehnungslehre" (1844), solving how to unify all boundary to interior relationships in orientable settings linear algebraically, hence enabling what today we call generalized Stokes.
Using Kannenberg's English translation (1995) Grassmann says in the preface of his book:
The initial incentive was provided by the consideration of negatives in geometry.
He gives an example of how sum of lines does not change if you allow for negative lengths as an example of his thinking then:
While I was pursuing the concept of the product in geometry as it had been established by my father*, I concluded that not only rectangles but also parallelograms in general may be regarded as products of an adjacent pair of their sides, provided one again interprets the product, not as the product of their lengths, but as that of the two displacements with their directions taken into account.
And he says that checking the previous idea that summation formulas still hold is indeed the case even then.
This new product we today call the exterior product and Grassmann notes that initially its anti-symmetry is perplexing, in modern notation $a\wedge b=-b\wedge a$. But this of course is precisely the keeping track of orientation.
Grassmann is aware of the benefits of this to the analytic setting and he remarks that Lagrange's work in analytic mechanics can be drastically simplified using this approach.
Grassmann at this stage is still emphasizing dimensions 1 through 3, describing how one can step up a dimension by keeping track of orientations. But he also states that "but in pure extension theory their number can be infinitely increased." Number her being dimension.
The reason why I wanted to add this here is the remarkable explicitness in Grassmann's tackling of orientation, which seems to me to be different to earlier cases of use of oriented areas. Orientation is not just a useful observation or property, it's the central concept that drives his discoveries.
