I thought that to make one non-polarized capacitor, the two negatives or the two positive leads from two capacitors have to be soldered together. Not solder the negative and positive leads together. Is this correct?
Right. (As was pointed out already.)
An electrolytic capacitor is a coil of metal for the positive plate, a chemical bath (connected to by the UNcorroded metal can) for the negative plate, and a layer of carefully cultivated corrosion between them for the thin insulation (dielectric) between the two, where "the capacitance happens". The corrosion layer is built by an electrochemical reaction driven by the positive charge on the plate, and during manufacture the voltage is started low (with a current limit - like a resistor) and gradually raised as the layer forms and thickens, able to take progressively higher voltages, until the design thickness is reached.
A can of liquid can have more than one foil plate wrapped up in it, of equal or differing sizes. This produces one capacitor for each plate, with separate positive and a common negative terminal.
If you charge it backward the chemical reaction cleans off the layer. Then the current electrolyzes the solution and the gas pops the seal and blasts the surroundings, and it's dead. Ditto of it's fed too high a voltage for the thickness of the layer and a substantial current flows.
When discharged the layer very gradually degrades. In use the layer is maintained by a little leakage current rebuilding any thin spots. You can leave it discharged (off) for months, but if it's discharged for years you have to work it up gradually. With an old tube radio from the attic you can do this over about a day with a variable transformer supplying lowered line voltageu, raised in several steps. (With modern semiconductor stuff you can do that to the raw caps on the supply, but if it's a regulated or switching supply the stuff downstream may go from no power to full voltage suddenly at some stage, so you might "blow off" some aged electrolytic on the boards.)
For AC you can make a "non-polarized" capacitor by putting two (identically sized) capacitors in series. At first they aren't charged and get cycled both backward and forward. But the leakage is mainly when they are charged backward, and this acts as a rectifier. Within a few cycles (and long before they build up a lot of pressure or damage the insulating layer) they develop a DC charge. They cycle between no-charge and double-voltage, rather than between forward-and-backward chage, and thus operate normally. The forward leakage currents rebuild the insulating layer much more effectively than the reverse currents tear it down, so it stays OK.
You can tie the negative pins and use the positive pins for the terminals of the resulting combined capacitor, or tie the positives and use the negatives. The usual approach is to tie the negatives, so the cans are tied together and don't need to be insulated from each other. A "non-polarized electrolytic" is one of those two-capacitors-in-one-can arrangements, with the central negative pin not brought and the can insulated.
You CAN put SLIGHTLY different caps in back-to-back series, but they divide the voltage unevenly, in inverse proportion to their capacitance. The smaller cap has to be the full voltage, the bigger one doesn't fully charge and may degrade as a result. But going too far may cause issues with the leakage-to-give-dc-charge bit, so it's better to just use two identical caps.
Plus-to-minus just give you a combined cap that's still polarized. Feed it AC and, maybe within minutes, the insulating layer breaks down, it generates gas, and jets chemicals or explodes.
