Why buckets have rims curled outwards? My brother  told me that household buckets have rims curled outwards to increase their polar moment of inertia.
But for what? This increased moment of inertia is needed to counteract what?
 A: 
Why have buckets with curved rims?

The polar moment of inertia is used to gauge the capacity of an object to resist twisting, or torsion, (in circular cross sectioned objects). In other words it is a measure of the angular displacement of the bucket when undergoing torque.
So if we filled the bucket with water, and then used the handle to lift it vertically, any slight horizontal motion would cause the bucket walls to twist, and reinforcement of the top by means of a curved rim would possibly reduce this twist, if the handle  pivot points  were located there on appropriately  reinforced areas.

Extract and image from Polar moment of Interia

A schematic showing how the polar moment of area is calculated for an arbitrary shape about an axis $o$.
$ρ$ is the radial distance to the element $dA$.
Definition of polar moment of inertia,
$${\displaystyle I_{z}=\int _{A}\rho ^{2}\,dA}$$
$I_z$ = the polar moment of area about the axis $z$
$dA$ = an elemental area
$ρ$ = the radial distance to the element $dA$ from the axis $z$
For a circular section with radius $r$:
$${\displaystyle I_{z}=\int _{0}^{2\pi }\int _{0}^{r}\rho ^{2}\rho \,d\rho \,d\phi ={\frac {\pi r^{4}}{2}}} $$

Two other advantages , with respect to plastic and other thin  walled buckets, should also be mentioned .
The curved rim  strengthens the plastic bucket and also increases the mass at the top, raising the center of gravity, making it easier to tip it over, (but only slightly).
The curved rim may reduce drips whilst pouring.
A: Moment of inertia? The difference of moment of inertia of a bucket with rim curled inwards and outwards is ridiculously small. But even if it was not -  nobody cares.
The rim is (usually) curled to reinforce the bucket and to make it safer (you would not cut your hand if the rim is curled).
The rim is curled outwards to make it possible to pour out all the water from the bucket without having to turn it completely upside-down.
