Agent 1 - I called on 3/13/2024 and spoke with one agent. He started the process of cancelling the flights and he was getting ready to charge me the $40 cancellation fee (as others have said, it appears this is the new way of doing things - you have to pay the cancellation fee first and then they issue the refund). I input my credit card details using the phone keypad, the charge was approved by my credit card company, but he comes back and tells me it didn't go through on his end. We tried again with the same card, same result. We tried a different card, same result. I insisted that the charges were all approved on my end, but he told me that I need to call my bank to resolve it. Since he had already had me on hold several times during the call, I asked him to wait while I called the credit card company. I called my credit card company and they said there were no issues with the charges and that there was nothing further they could do. While this was going on, the agent hung up on me. This call was 32 minutes long.

Agent 2 - I called back a few minutes later, got an agent that was in a call center and couldn't hear a thing, so hung up.

Agent 3 - I called back again and had to repeat everything over again even though I selected the option to continue with the same transaction that the VRU asks you about. We went through the same things except this agent tried sending me an email with a link to enter in my credit card info. The charge went through on my end, but he said it didn't go through on his end. He suggested I wait and call back in a couple of hours because it may be an issue with their system. I insisted that it shouldn't be this difficult to cancel a flight and get a refund and he responded that this is totally normal and that it's common for customers to have to call three or four times to cancel a flight. I told him this was ridiculous and asked if I could speak to a supervisor. He took down my contact information and said someone would get in touch with me. He then provided me with a case number. This call was 31 minutes long.

WhatsApp chat - from this forum, I found out that Iberia had a chat function via WhatsApp. Once I got through the bot, I got connected with an agent. I described what I wanted to do with the agent, she confirmed that it was $40 per passenger and asked if I wanted to still cancel the flights. I responded yes. She tells me the case has ben elevated to the responsible department, gave me a case number, and said that it takes 72 hours to 7 business days and that I would receive an email. She asked if the information was clear. I said no and asked what the next steps were. She said I just have to wait until I receive an email. The chat lasted 27 minutes. After this chat, the flights appeared to be cancelled. I couldn't access the reservation using the record locator anymore.

Received a call from a +34 number at 4:46 am. I wasn't awake to take the call, but I'm assuming it was Iberia calling back. No message was left.

WhatsApp chat - almost 7 business days had passed and I hadn't received anything (no email, no refund, no return of my avios), so I decided to try my luck. I got through the bot and got connected with an agent. I gave him all the info again, he didn't have any information on the status of my case and just said that within 24 hours I would hear something. I asked him why the cancellation can't be done right away and he didn't answer my question and just said that my case was escalated to the responsible department.

Twenty-four hours have passed and I still haven't seen any movement. No email, no call, no refund, and no return of my avios. I chatted again and got the same answer. They're working on my case, it's been escalated and I have to wait until they get in touch with me. No ETA on when that would occur.
I'll update this when I do hear back.

Decided to resort to Twitter to see if I could get any movement on this. Didn't make much progress, but the person manning the Twitter account said someone would get in touch with me soon.

Got a few calls from Iberia during the week, most of them were at 6 in the morning and I wasn't awake to answer. They did try one day at a normal hour, but I was in a meeting and couldn't answer. Finally, today, they called and I was able to answer. The agent basically told me the same thing, I needed to pay upfront to cancel and then the Avios and money would be returned. I insisted that it shouldn't be this difficult to cancel a flight and that I've been trying to cancel for two weeks. She confirmed some info, sent me an email to pay the cancellation fee, it seemed to work correctly, but no charge appeared on my credit card. The agent came back on the line and said due to all the inconvenience with cancelling the flights, they weren't going to charge me the cancellation me. The timeline they gave me for when my Avios and money would be refunded was something like 7 days. So, we'll see if this actually gets resolved soon.

Got a message from the social media folks on Twitter saying that the backend folks let them know that emails were sent out to us and that the Avios would be refunded w/in 72 hours and up to 30 days for the money to be refunded. I responded saying that I hadn't received any emails yet and they just told me to wait up to 72 hours.

Got the email saying that the Avios were back in my account and confirmed this by logging in.

Got an email from PayPal saying I was receiving a refund from Iberia.

Refund showed up on the credit card used w/ PayPal.