Two points. First is that though the TA doesn't change when adding Trichlor, the resulting lowering of pH DOES lower the carbonate alkalinity because of the drop in pH and it is that change in carbonate ion concentration that directly affects the calcite saturation index, though it is computed from the other factors of (mostly) pH and TA. So the reality is that you really aren't trying to target a TA level, but rather a carbonate alkalinity level since that (along with pH and calcium) is what is protecting plaster surfaces and preventing scaling (and also related to the rate of pH rise from carbon dioxide outgassing). This is all put together in the calcite saturation index. The Recommended Levels in the Pool School give rough ranges that have that index be (mostly) reasonable (exception being for SWCG pools it's too low at the low end and on average).
Second is that Dichlor is not the same as Trichlor and DOES change the TA by increasing it. This is shown below:
Dichlor Addition
Dichlor + Water ---> Cyanuric Acid (CYA) + Hypochlorous Acid (HOCl) + Sodium Hydroxide (NaOH)
Unlike Trichlor, Dichlor produces a strong base in addition to the two weak acids. A strong base raises the TA (as well as the pH). This is shown below:
........................
+TA
NaOH ---> Na
+ + OH
-
Sodium Hydroxide ---> Sodium Ion + Hydroxide Ion
The overall pH from Dichlor addition is close to pH neutral (only slightly acidic) because the strong base largely counteracts the two weak acids. As for the carbonate alkalinity, it only changes a little from the slightly lowered pH.
Sodium Hypochlorite (Chlorinating Liquid or Bleach) Addition
As for the hypochlorite sources of chlorine, they raise both the pH and TA upon addition. This is shown below for bleach or chlorinating liquid:
........................
+TA
NaOCl ---> Na
+ + OCl
-
Sodium Hypochlorite ---> Sodium Ion + Hypochlorite Ion
The pH rises because about half of the hypochlorite ion becomes hypochlorous acid so lowers the quantity of hydrogen ions:
-TA ..
+TA
OCl
- + H
+ <---> HOCl
Hypochlorite Ion + Hydrogen Ion <---> Hypochlorous Acid
Chlorine Usage/Consumption
Now all of the above is fine and dandy, but it neglects the fact that chlorine doesn't keep going up and up in the pool, but instead gets consumed/used and this is an acidic process that changes both the pH and the TA. More details about this are in
this post, but using the breakdown from sunlight as an example the following is what happens to TA:
...........................
-TA
2HOCl ---> O
2(g) + 2H
+ + 2Cl
-
Hypochlorous Acid ---> Oxygen Gas + Hydrogen Ion + Chloride Ion
+TA
2OCl
- ---> O
2(g) + 2Cl
-
Hypochlorite Ion ---> Oxygen Gas + Chloride Ion
So you can see that to the degree that the pH and TA rose from adding a hypochlorite source of chlorine because it produced hypochlorous acid, when that same hypochlorous acid gets used/consumed, the pH and TA drop back to where they started. Even for the hypochlorite ion that remained as is, it's addition increased the TA but when it gets used/consumed it decreases TA. The only net change is a small pH and TA increase from the "excess lye" in chlorinating liquid or bleach (I don't show that above).
The bottom line is that when you are trying to adjust your TA you do so by increasing it by adding some form of carbonates, usually sodium bicarbonate (baking soda), though using sodium carbonate (pH Up or washing soda) will also increase carbonates (but increases pH more). When you want to lower your TA, you do so by adding acid and increasing aeration of the water usually at lower pH. One should generally ignore the TA changes that occur when the pH is going up and down -- just focus on the TA level when you are at your desired pH.