First off, of course US presidential approval/disapproval ratings aren't exact complements of 100. Nor even are their changes 100% identical in magnitude except with reversed sign.
An approval rating of 40% can mean any disapproval rating from 0 to 60%, depending on share of those not expressing an opinion. For that matter, if you compute both #s on the basis of a set of polls, you might theoretically even get disapproval >60%, though it's highly unlikely
As a practical example, that article you linked to uses Huffington Post summary of Obama Job Approval
That summary has current Obama ratings as (Disapprove 50.2%; Approve 45.9%) - totalling 96.1%.
Having said that, there's an obvious correlation between changes in approval and disapproval (e.g., "No Opinion" in Gallup ranged from 4% to 7% entire 2015, meaning that approval and disapproval changes never shifted by a more than 3% magnitude difference between each other).
Just not a guaranteed identity in magnitude, as we can see from polls:
As an example, Gallup ratings for Obama recently had
DATE Approval Disapproval Appr_Change Disappr_Change
Jun 29 46 49 -1 0 (NOT "1"!!!)
Jun 22 47 49 +2 0 (NOT -2)
Jun 15 45 50 0 1 (NOT 0)
Jun 8 45 49 ....
Second, let's address your confusion from the article.
It stems from two independent isssues; I think you're combining imprecise thinking on the writer's part with a bit of mis-reading of what they meant.
Let's quote exactly what the article says:
the president’s approval rating today is still a significant predictor of which party will win next year. You can see that in the right-hand graph, which plots the popular vote for the same 16 presidential elections against the average net approval rating of the incumbent president recorded from June through September of the year before the election year — that is, right now.
In this graph, since we’re predicting the Democratic share of the vote, signs on approval ratings are reversed if the incumbent is a Republican.
First of all, as you can see from the exact quote, the article is NOT talking about calculating disapproval by negating approval rating. To be clear, the article never even MENTIONS disapproval ratings or the word "disapproval" - ONLY "approval".
They simply change the sign based on the party - because the value they are computing is "Democrat" share of popular vote, clearly, a higher approval of a Republican incumbent would have a negative effect on that where as a higher approval of a Democrat incumbent would have a positive effect.
Having said that, methodologically, their approach and formula seem somewhat wrong (or at least, woefully imprecise). Why? Two reasons:
The popular vote doesn't always sum up to 100% and is not a zero sum game. Just because having a popular (by X) incumbent Democrat would raise Democrat's share of the vote by F(X), that only means that having a popular (by X) incumbent Republican would raise Republican's share of popular vote by F(X). BUT, that doesn't guarantee that Democrats' share of popular vote would drop by the same F(X)!!! - it may very well shift the vote from undecideds instead of Democrat voters.
So, using "-F(X)" for Republican president is wrong for computing Democrat vote share, without proving that the impact is identical in magnitude on opposite party as on the same party.
Additionally, you also can't even assume that Democrats and Republican effects would be identical in the first place, unless you prove that.
So, not only is it wrong - as per above - to measure Democrat approval change by subtracting the effect of Republican approval change F(X) - but you can't even assume that Democrat incumbent's effect - let's call it F(D,X) - would be the same as Republican's F(R,X). They might be, but this article merely assumes it without proving it.
Correct formula would be (PVS=popular vote share):
PVS_change(D) = F(D,X)
PVS_change(R) = F(R,X)
F(D,X) = F(R,X) = generic F(X)
PVS_change(D) = - PVS_change(R)
Based on those assumptions, incorrectly arrived at over-simplified formula
PVS_change(D) = - F(X)
the exact chain of reasoning was:
PVS_change(D) = - PVS_change(R) = - F(R,X) = - F(X)