Why being invariant under the Lorentz transformation is so sacred for inertial observers?

I know that equations are invariant under the Lorentz transformation in SR, and thus if the lab observer measures that the moving observer's clock runs slower, the same is done from the viewpoint of the moving observer regarding the clock in the lab frame of reference. That is, in SR, there is no such thing as time shrinkage, and consequently, clocks never run faster from the standpoint of any observer.... Is it possible to have, contrary to the Lorentz transformation, an asymmetric system of equations for inertial frames as well?

Slower, faster or even at the same rate - that only depends on how an inertial observer conducts measurements and sets up laboratory equipment; for example, synchronizes clocks or, as in the example below, at what angle this observer turns his gaze.

Let's look into the famous 1905 Albert Einstein paper, § 7.

“From the equation for $omega‘ $ it follows that if an observer is moving with velocity $v$ relatively to an infinitely distant source of light of frequency $nu$, in such a way that the connecting line “source - observer” makes the angle $phi$ with the velocity of the observer referred to a system of coordinates which is at rest relatively to the source of light, the frequency $nu‘$ of the light perceived by the observer is given by the equation":

$$nu‘= nu frac {(1-cosphi cdot v/c)}{sqrt {1-v^2/c^2}}$$

This is Doppler’s principle for any velocities whatever.”

If an observer is moving towards or away from the source, there are longitudinal and transverse contributions into the relativistic Doppler effect. Hence, according to A. Einstein, at points of closest approach $(cosphi = 0)$ the moving observer will measure $gamma$ times higher frequency of light, or that the clock "at rest" is ticking $gamma$ times faster than his own.. If this observer was moving with velocity close to that of light, at this instant the "yellow" source of radiation would appear to him "violet", because all processes outside his spacecraft would appear to him as if in "fast forward" mode. This effect is known as transverse Doppler effect in source's frame, it is purely due to contribution of time dilation.

In this simple thought experiment - even in special relativity - two relatively moving observers measure non - reciprocal time dilation of each other clocks.

In rotating frame, for example, an observer on a rim of rotating ring simple cannot ascribe himself a state of "rest". If he will look (at a source of radiation) into the center of circumference at right angle he will see nothing then. This (rotating) observer can only see blueshifted frequency, nothing else. This observer is not also able synchronize clocks on a rim of rotating disc Einstein - way, because this synchronization along the whole rim gives non - vanishing time difference.

It should be noted, that if an observer in rotating laboratory synchronizes a pair of clocks in his lab Einstein - way, measured by these clocks rate of a clock in the center of the circumference would appear to him as running slower; but, again this synchronization won't work along the whole rim.

For the sake of convenience and simplicity of equations any inertial laboratory can be considered as a "stationary" one, however, of course this is not the only choice.