Be very careful about this topic and check the sources of any info you get. While there are some interchangeable cartridges, there are also some that will chamber but that are very dangerous.
There aren't to my knowledge any modern centerfire rifle cartridges that are "interchangeable."
There are a number of cartridges that have multiple different names, for instance the 6mm Remington is also the .244 Remington. Older Model 700's are marked .244 Remington but Rem changed the designation (and the rate of twist) to 6MM because the original rifles wouldn't stabilize the longer bullets and also because of competition with the Winchester .243. Others that come to mind include the 30-30 Winchester which is also known as the .30 Winchester, .30 Marlin, .30 Savage and the .30 W.C.F. All different names used by different manufacturers for the same cartridge. There are several other older cartridges in this category.
You mentioned .223 and 5.56, these are different cartridges and many (most) rifle manufacturers that make .223's will tell you not to shoot 5.56 ammo in their guns. However, .223 ammo is safe to shoot in a rifle marked 5.56 (lots of articles about this out there)
.22 rimfires are a complicated genre but without getting too far into the details, in a rifle marked .22 Long Rifle (.22 LR) you can shoot .22 BB caps, .22 CB caps, .22 Short Blanks, .22 Shorts, .22 Longs, and .22 Long Rifles. The shorter stuff may not feed properly or operate semi automatic actions, but they will fire safely. Doesn't make it a good idea necessarily, but it wouldn't be unsafe.
Handgun cartridges are a different story.
There are many examples of interchangeable handgun cartridges. For instance, you can safely shoot .38 Short Colt, .38 Long Colt and .38 Specials in a gun marked .38 Special, and you can shoot all of these plus .38 Special +Ps and .357 Magnums in a gun marked .357 Magnum.
While this is an interesting topic, again, be very careful. There are a number of very dangerous combinations where you could chamber and fire a cartridge from the wrong caliber and have a catastrophic result. Examples include .308 ammo in a .270, .7mm Weatherby Magnum in a .7mm Remington Magnum etc. Here is the SAMMI list of these dangerous combinations.
http://saami.org/specifications_and...1-Unsafe_Arms_and_Ammunition_Combinations.pdf