If you break apart these two sections then you have two independent 1st order filters. And if you mathematically combine them without taking into account loading, then swapping them around doesn't change anything.
For example, your left section is \$\frac{s}{s+\omega_{_0}}\$, where \$\omega_{_0}=\frac1{R_1\, C_1}\$, and your right section is \$\frac{\omega_{_1}}{s+\omega_{_1}}\$, where \$\omega_{_1}=\frac1{R_2\, C_2}\$. You can just multiply those to get \$\frac{s}{s+\omega_{_0}}\cdot\frac{\omega_{_1}}{s+\omega_{_1}}\$ for the total transfer function, if one isn't loading down the other one. And since multiplication is commutative, it doesn't matter which one goes first.
However, one does load down the other.
In either combination you have a combined 2nd order filter with three important parameters: gain \$A\$, frequency \$\omega\$, and damping factor \$\zeta\$ (or quality factor where \$Q=\frac1{2\,\zeta}\$.) Changing the order modifies \$A\$ and \$\zeta\$.
But it does not alter \$\omega=\frac1{\sqrt{R_1\,R_2\,C_1\,C_2}}\$, which is the same for both cases.
As shown above, you have \$A=\frac1{1+\frac{C_2}{C_1}\left(1+\frac{R_2}{R_1}\right)}\$. Flipped sections provide \$A=\frac1{1+\frac{R_2}{R_1}\left(1+\frac{C_2}{C_1}\right)}\$.
For both cases, as shown above or sections flipped, \$\zeta=\frac1{A}\cdot\frac12\cdot R_1\,C_1\cdot\omega\$. But since \$A\$ is likely different between the two cases then so is \$\zeta\$ also likely to be different, now.
The above discussion doesn't assume that \$\omega_{_0}=\omega_{_1}\$, so they can be different and the above math still applies.
But a further simplification comes if you do assume \$\omega_{_0}=\omega_{_1}\$. Then you know that \$R_1\,C_1=R_2\,C_2\$.
Then as shown above you have \$A=\frac1{2+\frac{C_2}{C_1}}\$. And with the sections flipped you have \$A=\frac1{2+\frac{R_2}{R_1}}\$. So, as in the case shown above, \$C_2\ll C_1\$ then \$A\approx \frac12\$. And when the sections are flipped, if it is also the case that \$R_2\ll R_1\$ then also \$A\approx \frac12\$. (With your values above, the flipped case would fail the \$R_2\ll R_1\$ test, though.)
That captures the idea where the 2nd section doesn't load down the first section. \$A\$ will be about the same value in both cases when their associated tests succeed, and then \$\zeta\$, which depends on \$A\$ will also be about the same.
There's two remaining ideas, I suppose. One is the input impedance of the total filter. The other is the output impedance of the total filter. And these matter with respect to a non-ideal source and a non-ideal load.