UK tourist visa for Americans 180 days in / 60 days in

There is no "180 days in." There is no "30 days out."

As a US citizen visiting the UK, you normally will not have a visa. Instead, you are a visa-exempt visitor. You are normally admitted for six months, not 180 days. You can enter for up to six months each time you enter. There is no minimum period outside the UK before you can enter again. For example, you can in theory take a short trip to Paris and return to the UK later the same day, the next day, or any time thereafter.

However, UK border officials are very diligent about the requirement that visitors be "genuine visitors." If you have been present in the UK for close to six months, and you leave the UK for a short period, you are likely to face more probing questioning on your return than you did the first time. You may be taken to "secondary inspection" where officers may interview you for as long as several hours before making a decision. Of course, it is also possible that none of this will happen.

The officers will consider whether you are trying to live in the UK through successive visits, which is not permitted. They will investigate whether you have been working in the UK or are likely to do so, which is also forbidden (this includes working remotely for an employer outside the UK). They may decide to admit you for six months or for a more limited period, which may be as short as a few days, depending on your circumstances.

It seems from the title of the question that you are considering visiting the UK for 180 days and then for 60 days. This might be possible, but you'll probably be more likely to succeed if the reason for wanting to stay for six months the first time, and the reason for wanting to come back after the initial six-month visit, are unusual.

In fact, when you enter the first time, you are likely to face a higher level of scrutiny when you tell the officer that you're planning to stay for 180 days. As noted in a comment, you can probably avoid this by using an ePassport gate. If you use a gate on your return, you might get lucky and be admitted directly by the gate, but you might also be flagged for an interview with an officer. The algorithm used to select visitors for such interviews is, naturally, secret. There is some more detailed but slightly outdated guidance about the gates at the website of the registered traveler service.

If you do manage to stay for so long in the UK, you will want to consider whether your extended presence there has any effect on your tax liability in the UK. This is a major consideration for Canadians, typically retired Canadians, who spend a lot of time in the US.