Can I put cathedral-port heads on a 6.2 or rectangle-port heads on a 5.3?
Yes, the heads physically bolt on, but the intake and the bore have to match, and that is where most of these swaps go wrong. LS heads come in two intake port shapes: cathedral and rectangle. The intake manifold has to match the head, and the combustion chamber has to match the bore and the piston.
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Cathedral-port castings: 241 (67 cc), 243 (64 cc), 799 (64 cc), 706 (61 cc), 862 (62 cc) and 317 (71 cc). The 243 and 799 are the same casting and the same 64 cc chamber; the 799 is the truck version, usually without the LS6-spec springs and lightweight valves. The 706 and 862 are the small-chamber 4.8/5.3 truck heads. The 317 is the common iron-block 6.0 (LQ4) head with a large 71 cc chamber.
Rectangle-port castings: 823 (LS3/L92, 68 cc) and 821 (also rectangle port; its factory application is listed two ways, LS3/L92 or the supercharged LS9/LSA). Rectangle heads flow more than stock cathedral heads and need a rectangle-port intake such as an LS3/L92 manifold. A cathedral intake will not seal on a rectangle head, and a rectangle intake will not seal on a cathedral head.
Cathedral heads on a 6.2. A 6.2 LS3-family block has a 4.065 in. bore, so a cathedral head with a chamber sized for that bore can physically fit. The questions are chamber volume, piston valve relief, pushrod length and the intake. A 243/799-style cathedral head raises compression compared with the 68 cc rectangle chamber, and you need a cathedral intake that fits the chassis. It is a combination to check, not a straight bolt-on.
Rectangle heads on a 5.3. A 5.3 has a 3.78 in. bore; every 6.0 is 4.000 in., and the 823 chamber is sized for the larger 4.065 in. LS3/6.2 bore. The head bolts on, but the chamber can overhang the smaller bore and the valve relief in the piston may not match. Many builders avoid this combination, and the ones who run it check bore clearance and piston-to-valve clearance first. It is not a cheap way to get LS3 flow on a 5.3.
Chamber size changes compression. A smaller chamber raises compression: 706 at 61 cc and 862 at 62 cc are the high-compression truck choices, 243/799 at 64 cc are in the middle, 241 at 67 cc and 823 at 68 cc are larger, and 317 at 71 cc lowers compression on a 6.0. Match the chamber to the fuel and the piston, not just to the flow number.
What to check before buying heads: the casting number, the port shape, the chamber volume, the bore the head was made for, the intake that matches it, the valve springs for your cam, and the piston valve relief. If any one of those is wrong, the heads are not a bolt-on.
More: LS heads and intakes guide · 243 vs 799 vs 706 vs 317 · LY6 vs L96 vs LQ9.