USB Power Delivery over Type-A connector

I have a Xiaomi phone that came with a 33W charger. It says it does 5, 9, 11, 12 and 20 volt, so looks like a fairly normal USB-PD charger. Except, it has a type-A socket and came with an A-to-C cable! How is that possible?

Seems pretty apparent to me what they did, they violated the USB spec.

A USB-A port by the USB spec is not to provide more than 5 volts and more than 3 amps. It may be no more than 2.4 amps depending on which spec applies. There's a little bit of room +/- on the 5 volt spec so seeing 5.2 volts will be common, perhaps 5.5 volts might still pass inspection, but 9 volts is just right out. I remember a big deal a few years ago about big names in tech cracking down on nonstandard USB chargers, Amazon and Apple pulling noncompliant products from their stores, Google and Apple threatening to pull any licensing agreements for putting so much as 6 volts on their USB chargers. This is overboard on the breaking of specs like I've never seen.

In looking for some images of this charger on the internet I can't know if the photos I'm seeing are of the same item you have, but what I do see is quite telling. There's no UL listing or other industry safety markings. Nowhere do I see "USB-PD" on the product or in the description, there is no indication that this is a product that even attempts to meet the USB specifications.

It seems to be doing normal PD on the C end, because I successfully charged a laptop at 20V. Both charger and cable have the 5 extra pins from USB 3. Is there a standard for this at all?

No. The USB-A connector was never intended to handle more than 12 watts, 5 volt @ 2.4 amps. It wasn't intended to handle more than 5 watts actually but with some redesign and exploitation of some over engineering in the original design they decided that 12 watts was safe.

Does the A-to-C cable have a non-standard pin mapping to make USB-PD work?

I'd say that is quite likely. Putting voltage on something in the air has the potential to arc. There's ways to minimize this and deal with it if it is unavoidable. Also of concern in a connector that carries power is the heat it will produce from the inherent resistance. One way to lower the heat is to spread out the area of contact. The contacts in a USB-A connector can be only so big and still fit other USB-A plugs. What can also spread out the heat is use more contacts. I have no doubt that they used contacts intended for data to transfer power.

USB-BC, the old "battery charger" spec uses the D+ and D- pins to communicate how much power the connected device was safe to draw. In the pre-BC days USB power supplies would leave these pins unconnected and devices were just expected to behave themselves. Then came all the crazy ways people tried to get more power through USB that got a lot of people upset. Consumers were upset because USB chargers were slow and when they got faster then they were no longer compatible with each other. Some consumers even saw their electronics go up in smoke.

Apple, Google, Amazon, and others were upset because if people can't trust their USB devices from working with each other, and possibly starting on fire, then they had a possible public relations nightmare on their hands. To combat this came USB-BC to push the power allowed to a more acceptable 12 watts, and a new standard on communicating how much power a device could safely draw from a power supply. Not long after came USB-C and USB-PD to allow for more powerful devices to charge from a standardized port.

It just boggles the mind why people go through the effort to engineer these strange and unsafe variations on existing themes. I could have some understanding why this was done in the days before USB-BC and USB-PD. At that time there was a government mandate for phone manufacturers to decide on a common connector for power, this was from China, the EU, or both. The industry chose USB. That lasted about 2 seconds before phone manufacturers started to violate the standard in horribly unsafe ways. The industry moving to USB-C and USB-PD should have removed even the desire to do anything else.

But here we are. Still seeing USB power bricks that can't play by the rules.

If that were my USB power brick then I'd be tempted to smash it to pieces and throw out the bits. That has the potential to damage standards compliant hardware with its non-standard power output. It is not safe and if you choose to keep it then handle it with care. Use it only with the cable and phone of the same brand.