Replacing receptacles - why are white and black wires crossed?
Check the tabs.

Check the tabs.  That is the #1 trouble spot.  If a tab is broken off on the old receptacle, do the same with the new.

When there are 2 wires on a side, what that means is totally different if the tabs are broken off, versus not.  

White wires on silver screws, black on brass screws

The old one should be like that.  If you see a case where it is not, take some good photos of the back of the box and both sides of the receptacle, and ask a new question.   Because there might be something special going on there. 

What you call "crossing" should simply be the wires getting to where they belong.  

If you have been disregarding this, and "uncrossing" the wires, then you need to go back and correct that.  In those cases the outlet will probably function normally, but will create an electrocution risk situation. 

If the tabs are not broken off, then the wires simply need to obey the above rule: you can swap two whites on two silver screws if that fits better.   

If the tabs are broken off, then position also matters. 

While you're in there, ground the receptacle

The one in your photo is grounded.  If you find any that are not, get some #12 bare wire, and some wire nuts, and run a pigtail from the group of grounds to the receptacle.  

If you can't ground, then have a conversation with someone knowledgeable about GFCI protection (don't simply slap in a GFCI receptacle in every location; but if you insist on doing so, only use the LINE terminals and leave the warning tape on the LOAD terminals). 

On switches, screw color is vital, and wire color is unhelpful.

I gather you'll be doing switches too?  In switches, wire colors are almost meaningless.  In 1-way switches (2 screws only), it doesn't matter.  However, 3-way and 4-way switches absolutely require care in matching wires to screw colors.  I like to mark travelers with yellow tape.  Travelers are on yellow screws in 3-ways. On 4-ways, one pair is on yellow, the other pair is on black.