Time dilation depends only on the magnitude of relative velocity, not direction. That is captured in the Loedel Diagram. It is a Minkowski diagram with each frame moving in opposite directions:

The power of this diagram is that it shows the paradoxical symmetry of time-dilation, and from a Newtonian mindset, it is paradoxical, but in relativity, as the observers change distance, the bias defining their "clock synchronization" changes in manner that allows each observer to consistently say the other's clock ticks at $1/\gamma$. We'll come back to it.
The linked Ted-Ed video on the twin paradox only explains what single local observers see while looking out their windows, and what they see is a doppler shifted signal:
$$ f_{\pm} = f \sqrt{\frac{1 \pm \beta}{1 \mp \beta}} $$
(Note that the radical sign brings in a important symmetry that does appear in the sound-wave doppler shift...a symmetry that is broken by the existence of an aether, or stationary oscillating medium.)
The standard figure is shown:

As seen in the figure, traveling twin see half $f_-$ followed by half $f_+$, while Earth twin sees lots of $f_-$ followed by a short period of $f_+$, and it works out so that the total number of pulses seen by Earth is a factor
$$\gamma = \frac 1 {\sqrt{1-\beta^2}}$$
fewer than those seen by Space twin.
The does not really explain the Twin Paradox, because as you point out: each observer sees the other twin's clock moving slower on both inertial legs of the trip. To resolve this, we do not want to consider what the twins "see", meaning physically see looking out the window, as that is corrupted by Doppler shifts.
Rather, we consider each twin to have access to a co-moving lattice of synchronized clocks (and rulers) with which they can record and reconstruct the space and time at any moment along their trip.
Defining $T=L/\beta$ to be the time it takes for 1 leg of the trip, then what is measured is:
- Earth measures Earth clock ticking: $T+T = 2T$
- Earth measures Space clock ticking: $T/\gamma+T/\gamma = 2T/\gamma$
- Space measures Space clock ticking: $T/\gamma+T/\gamma = 2T/\gamma$
- Space measures Earth clock ticking: $T/\gamma^2+T/\gamma^2 = 2T/\gamma^2$
meaning space twin has "missing" time:
$$ \Delta T = 2T - 2T/\gamma^2 = 2T(1-(1-\beta^2))=2T\beta^2 =2L\beta $$
What happened?
Well, when space twin turned around, he needed a completely new lattice of co-moving observers, as his definition of "now" at Earth jumped forward by $ \Delta T$, as shown:

The line of simultaneity at turn around changes from the top blue line to the bottom red line. Note that space twin does not see Earth's clock running faster, rather, it is a recalibration of the bias required to synchronize the clocks. (It is reversible, if the twin turns around again, then he goes back to the blue lines and the time on Earth has now gone backwards, which is a perilous situation for time-dilation, but it completely acceptable for clock bias).
That is the resolution of the twin paradox; however, the result remains paradoxical. It means at the turn around point (or event), the time on Earth is not defined. Depending on velocity, it can vary by $\pm L/c$. What that means is the right now, 12/31/20, the time on Alpha Centari (4.37 ly away) can be anytime between 8/19/16 and 5/14/25.
That peculiarity of relativity is known as the Andromeda Paradox, and leads to 4-dimensionalism, which is truly paradoxical.