Do (simple, inorganic) salts ionize completely in water when they dissolve? Specifically: Suppose I add equal parts (i.e. the same number of moles of each) sodium bicarbonate and potassium sulfate into a container of distilled water large enough to dissolve them completely. In a second container of water, I add potassium bicarbonate and sodium sulfate. Are these two containers now physically identical? (That is, is there any experiment that could tell them apart?) My understanding is that each now contains free-floating Na+, K+, S-, and HCO3- ions. And if I evaporate away all the water, what will be left behind in each container?

If I replace the potassium and/or sodium with, say, magnesium or calcium, and the bicarbonate and/or sulfate with chloride or citrate, do the same principles hold? (Assuming the resulting salts are water-soluble - some, like calcium carbonate, won't be.)

(I'm not sure if the list of cations and anions above have much in common chemically. My interest relates to the fact that they can be added to water to simulate the chemical makeup and flavor of natural mineral spring water.)

(PS: What's a more elegant way to say "the same number of moles of each?" If I say to mix things in equal parts, or in some ratio, I would normally specify whether I mean by mass, by volume or... how do I say "by number of moles?")
When a salt, no matter which salt, or any other polar compound is exposed to enough water, then its polarized molecule parts, depending on its polarity, become attracted to the oxygen or hydrogens of the water, which will result, starting with the outermost molecule layer, in the complete dissolution of the compound. Within this process both the positively and the negatively polarized sections will get surrounded by water molecules (= hydration shell) and thus shielded from one another. In this state the original compound(s) have (theoretically) disappeared completely and the two containers will indeed be identical. Possible back reactions will (mainly) result in the most energy efficient compound for the particular experimental conditions and have the same chance in both containers.

When you now evaporate all the water you may get various compounds and mixtures depending on what was dissolved.  

PS: I'd say "of equal molarity" or "they are equimolar", but my mother tongue isn't English, so there might be a more elegant way.