In lay terms, what are the real world consequence of the gauge invariances/symmetries upon which the Standard Model is built? We learn that the SM is based on gauge invariance. Gauge invariance in turn is a consequence of symmetries (as I understand it) - meaning that a gauge theory having a symmetry is what makes it a gauge invariant theory? (If wrong, please do correct!)
But these gauge theories and symmetries seem to imply actual physical observed or deduced findings - mathematical counterparts of conservation laws, and symmetries, observed in the physical universe, to which particles and interactions appear to adhere, and from which enough structure could be deduced for such a model to develop.
(I'd also imagine that once a structure is posited, based on some symmetries, further symmetries may become predicted and then searched for to complete and validate the model, much as happened with elements and the Periodic Table in chemistry.)
I'm unsure which is chicken and which is egg - the items/properties/measurements which are gauge invariant, or the symmetries we observe that allow us to fix a gauge for these and lead to gauge invariance as a property of the theory.
But the $SU(3)\times SU(2) \times U(1)$ structure, and its local/global gauge symmetries, presumably do correspond to real world findings that appear to have real-world measurable corroboration/meaning. Or at least presumably some should, otherwise falsifiability becomes an issue.
I'm curious what those would be, so far as they are readily explicable or indeed enumerable. Or the main ones for the fields within SM, that specifically lead to the SM structure being the structure of that model. It would help deepen my admittedly very amateur understanding of the SM.
 A: It was the gauge invariance that was discovered first - in fact in Maxwells electromagnetism in the mid 1800s - that this was also a symmetry was understood later. In fact, that this was a symmetry was understood implicitly immediately but it was seen as a curiosity rather than a principle upon which physics could be built.
Weyl suggested soon after the discovery of GR that gravity and electromagnetism could be unified if a local variance of scale (gauge) was taken into account. After the discovery of QM, Fock and London modified Weyl's suggestion by locally varying the complex phase (which is a U(1) symmetry) and this explained the effect of electromagnetism on the wave function of a charged particle. This was the first gauge theory in the modern sense. Whilst this has a U(1) symmetry it is not the U(1) symmetry of the Standard Model.
It is the charges of the symmetries that are most directly physically relevant. For example, in the SM we have that the charges of:

*

*The U(1) symmetry is weak hypercharge


*The SU(2) symmetry is weak isospin


*The SU(3) symmetry is colour
The abstract symmetry for weak isospin is $U(1)$, this is a circle. This is called the gauge structure group. (Actually physicists call this and another much bigger group the gahve group. Whilst mathematicians call the former the structure group and only the latter group, gauge group. This is why I have opted for the preceding terminology).
The correct mathematical technology to think of this is the technology of fibre bundles.  Physically, think of every point of spacetime having an attached circle and a charge - the weak isospin -  running around it. Ditto for the others. Though it is a bit harder to see how a charge runs around for SU(2) and SU(3) and whose charges are 3d and 8d respectively unlile the case of U(1), the circle, where the charge is 1d.
