I remember when $10 would buy a bag of groceries with change. $20 would buy several bags of groceries.
Ones and fives were the most common bills for change. It made sense they were in the right side of the cash drawer.
We use bigger bills today but the cash drawers haven’t changed.
I used to live in Westchester. I can assure you that they do do everything backwards there and have strange names for things. I really hated everything about that place. I live somewhere else now 
What years are you actually referring to here? Cash registers have been “doing the math” for at least three decades, I’m sure. Are you suggesting that the OP’s jobs were before then?
Every place I worked had the lower denominations on the left.
I don’t think I’ve even seen anyplace that kept the lower denominations on the left. I’ve always put them on the right, and seeing the opposite should be something that would stand out.
What is this!?!?:eek:
Not everyone does it the correct way of credit card, void out and refund receipts, 2, 50 and 100 dollar bills on the far left then 20 10 5 1 bills and 50 cent/dollar coins followed by quarters dimes nickels pennies also from left to right?
It’s been a lot of years since I was responsible for a till, but I still remember the explanation for it. It’s how we write our numbers and do math, higher values to the left. This was right before registers became work stations in a network with touch screens. The register would do the math, but if you weren’t proficient at keying in the amount tendered or miskeyed, you had to be able to do the math in your head. Especially if you miskeyed, because once you hit the sale button, everything but the incorrect change was gone without looking at the tape or receipt.
Except largest on the left, smallest on the right precedes the days of electronic cash registers. And when I worked a register, I could count back change without the register telling me how much to give back. As others have said, most people are right-handed and use their right hand to take change from the drawer, so it’s easiest to start with the smaller denominations on the right.
Every place I ever worked with a cash drawer , including a bank, had the larger bills on the left . And none of those places were in the Bronx or Westchester or even Long Island. They were in Queens, Brooklyn and Manhattan. I could count back change from right to left ( and sometimes had to even when the register did the math - I wasn’t going to void he whole thing because I mis-keyed the amount tendered,
Whether the higher bills are on the right or left is arbitrary, just like driving on the right or left side of the road. It doesn't matter if we drive on the left or on the right- what matters is that a particular place chooses one or the other and sticks with it. But for the cash drawer, consistency doesn't matter- there's no reason that McDonald's has to do it the same way as Wendy's which is he same way as Dunkin Donuts.
I’ve worked retail in a candy store, two shoe repair stores, and now a big box franchise with 240+ stores in six states. All of them were large denominations on the left, smaller on the right.
Really, the most important thing is for the store to be consistent to minimize fumbling and giving out the wrong change,
Australian hotel.
Bills are in a row across the top, lower denomination to the right.
Coins are in a row across the bottom, higher to the right
Something like this, where “(” and “)” are the till edges, and “|” are the dividers.
(cc slip|50$|20$|10$|5$)
(5¢|10¢|20¢|50¢|1$|2$)
$2, $50 and $100 bills go under the tray so someone can’t just reach in and snatch them. Or, if you have one of those safes-with-the-slot under the counter, they go in there immediately–after you gold pen them; they’re checking $10 bills, now.
The manager of a Blockbuster I worked at demanded all bills had to face the same way–three strikes and you were out. Then he got fired for stealing.
This is why no one uses two-dollar bills. Cashiers hide them away instead of giving them out as change.
I doubt anyone is going to be especially tempted to snatch away a two-dollar bill.
Sorry, I meant $2 bills went under so you didn’t give back too much change (if you stuck it the ones slot.)
You’d be surprised how many people refused to accept a $2 bill as change.
Another “cash registers I’ve used have big bills on the left, small ones on the right” vote here.
I’ve never been given a two-dollar bill as change, except once when I specifically asked for one. Several times, upon asking, I was told they weren’t kept in the drawer.
After reading the OP, I paid attention today for one datapoint. The one French bar I visited today had the larger denominations on the left.
At the museum where I work, smallest to largest goes right to left.
That’s the only till I can recall ever operating, so I have no comparison to work with.