Neutralization reactions are reversible. In theory, at least, even if not so much in practice, all reactions are reversible.
If you look at things on a molecular scale, there's a principle called microscopic reversibility. In effect, there's no physical reaction mechanism you can create that's a one-way door. Maxwell's demon doesn't exist, and you can't get around that no matter how many steps you chain in whatever order, or which particular reagents you use.
What you can do, though, is set things up so it's exceedingly unlikely that the reaction proceeds in the other direction. That's what's happening with reactions we normally deem "irreversible". There's something about them which makes it difficult, in practice, for products to go back to reactants.
One common feature is the size of the equilibrium constant. If a particular reaction has an equilibrium constant that's really big, the equilibrium for that reaction is going to be vastly tipped to the product side. So much so, that the reactant side of the reaction will not be present in appreciable amounts. (For example, this is why combustion is viewed as irreversible. The reaction is so favorable toward the products that only parts per million or parts per billion of the reactants will be present at equilibrium.)
Another method for promoting "irreversibility" is for the products to partition to another phase. For example, if the reaction forms a gas or a precipitate, that product is removed from the reaction volume, and the concentration of the products drops, pulling the reaction toward the products. Note that even in this multi-stage setup, each step is reversible. That evolved carbon dioxide molecule could re-dissolve in solution, and then react backwards, but it's terribly unlikely for it to do so. You've basically fixed one of the products at a very low concentration, which draws the equilibrium toward the product side.
Precipitates work much the same way, as the activity of the solid is fixed by surface behavior, and does not increase because you increase the amount of solid you add. (The rate of reaction may increase by adding more solid, but the equilibrium of the reaction does not.)
So in your examples, the reason you get hydrolysis or neutralization is due to how you set up the reaction conditions. Have a large excess of HCl? You drive things toward neutralization. Have a way of getting rid of HCl (for example by driving it off as a gas, or reacting it with a base)? You drive things toward hydrolysis.