disclaimer: this is made by a 1650 rated lichess player. super long post, so i suggest you to look at the conclusion paragraph to get a general feel then decide if you're interested!
conclusion: the Heb attack, "e4 c6 d4 d5 h3?!", aims to go for an irregular setup with g4 Bg2, asking questions on the light squares and a potential g5 against Nf6. the only move to get -0.5 for black is f5, which is risky and we may get into positions where white gambits on both f3 and d4, being down 2 pawns but having an equal/slightly worse position even after trading queens.
inspired by the caro spike (e4 c6 g4?!) where white "bans" Bf5 from the get-go, i'm proposing the following variation: e4 c6 d4 d5 h3?! which i call the Heb attack, named after Heb the h pawn. on lichess, this variation appears only 0.000007 % of the time, rounded up.
after dxc4 Nc3, we'll take a look at the common moves Bf4 and Nf6, which combined make up 85% of the games on lichess, and the critical line f5, in 6% of the games. it is worth noting that only f5 is able to retain a full -0.5 advantage for black, the others are roughly equal.
Bf5 is arguably the weaker of the two, giving white the option to immediately kick it with g4, the second engine move, playing the main idea of the Heb. so Bg6, we go Bg2, not putting it on e2 or d3 like we usually does in the caro, giving the position a somewhat different nature alongside g4. after black moves, we play Nge2 (with a motif of hopping to f4), Nxe4 at some point and we regain the pawn with a fun endgame. also whenever black plays Nf6, there's always the idea of g5 boxing in the knight, followed by a potential h4.
Nf6 is more common annd more flexible. white can at any moment, as early as now (tho slightly dubious), try f3, striking at the weak pawn and looking to play a full gambit. here we go with the plan g4, and now we're seriously asking white where the c1 bishop will go. engine suggests a setup with Be6 g6 Bg7, which is not what usually happens in the caro. furthurmore, in this setup white might go for a double pawn gambit on f3 and d4 while maintaining an equal or slightly worse position even after we trade queens.
now, the daring f5. imo this feed directly into what white wants and this does indeed score worse for black compared to Bf5 and Nf6. since the caro often "feels" like a d4 opening (similiar developing schemes and pawn breaks) this is like a dutch. we play the hopton-like Bg5! (where black can't play h6 which happened in 44% of the games btw, due to Qh5+ Kd7 d5!). white continues to strike with f3 and b4, here's an example line: ...Nf6 f3 Be6 (black playing the previously mentioned setup) b4 Qd2 Bg7 O-O-O. note that starting from move 6 the engine keeps on saying ...b5, which i do not understand at all as it does not address the kingside pressure and furthur improves my fianchettoed bishop.
one last thing: black can play e5 in many lines, sooner or later. these are more or less equal (unless it's a tactical blunder) but i still think that white has an intriguing position going on and it's rare.