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In particular, if I cut a block of stainless steel in half, would the newly formed faces be stainless as well?

It is usually a bulk property though you would need to know exact regulations for your country to be certain.

Stainless steel is steel (i.e. iron + a little bit of carbon) melted with another metal which makes it resistant to oxidation by atmospheric oxygen (usually chromium). It does not mean though that it would resist to strong acids (e.g. concentrated hydrochloric acid) or strong oxidizers.

Let's divide the steel world into two classes: 1) rusting steel and 2) stainless steel.

Rusting steel, in the presence of oxygen and moisture, will oxidize, forming hydrated iron oxides/hydroxides which have a greater volume than the original iron, and which have relatively little adhesion to the metal. They curl up and continue to expose bare metal, and so rusting iron/steel will continue to rust.

It was discovered in the early 1900's that addition of 12% or more chromium to iron would produce an alloy that oxidized very slowly, forming a very thin oxide layer that was adherent and did not continue to thicken. It resembled chromium in that respect, and the oxide was probably rich in chromium. Nickel was also found to increase the corrosion resistance of the alloy.

Now the bulk composition affects the surface oxide, but it is the surface that stains, or rusts - or doesn't. If you broke a block of stainless steel in half, in a vacuum, the fresh surfaces would be bare metal, active to many reagents. Exposure of the fresh surfaces to oxygen will oxidize them fairly rapidly (minutes, hours) to a passive state which resists many chemicals, like acids, especially oxidizing acids like nitric. (Interestingly, plain iron dissolves in dilute HNO3, but in concentrated HNO3 (>30%), the oxidizing power of the acid is so great that the surface of the iron is rendered passive: an oxide coat is produced which does not flake off and does not dissolve in the acid.)

Stainless steels are less resistant to chloride ion, which has a way to infiltrate the oxide layer and corrode the base metal. Stainless steels which have been brushed with a wire wheel made of ordinary steel will have a surface contaminated by tiny fragments of regular steel, which will rust. The oxide layer on the stainless steel body will be imperfect, and oxidation will progress thru and under the passive oxide, until the whole stainless steel is corroded.

Since it is the surface of the steel which contacts the active reagent, it would seem that the surface (oxide) is the determining factor, but, of course, the bulk composition strongly affects the surface oxide layer. And the surface needs oxygen to resist further corrosion!

"Stainless" is not a specific definition. The stainless steel with the least alloy is $5% ; ceCr$ according to AISI (It can't be cut with an oxygen/acetylene torch like regular steel). API considers $ceCr :Mo$ (9:1) as stainless for oil well tubulars. SAE consider $12% ; ceCr$ as stainless (most modern auto exhaust pipe). Stainless cutlery is also $12$ to $13% ; ceCr$. Non-magnetic stainless starts at the proverbial $ceCr :Ni$ (18:8) (Grade 304 and a half dozen other grades). All will resist corrosion in some environments and corrode in others. And, you can make a pretty good income telling people which stainless they need in their specific environment.