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I recently had an electrician come out and replace my electric panel.

He seemed very knowledgeable and overall appears to have done a good job, even went through and labelled most of the panel... Except for three breakers at the bottom of the panel.

One of these breakers is for the pool lights and an outlet near the pool, so he put a GFCI on it and it works fine, but it wasn't labelled...

The other two breakers above it, I found out, are actually two single pole breakers that I believe are both used for a pool pump that uses 240 V, and it should have a double pole breaker installed.

I think one of these single pole breakers is providing one leg of the circuit and the other breaker is providing the other leg...

This seems unsafe to me. If someone were to mistakenly leave one of these two breakers off, but the other on, the pool pump definitely doesn't run, but it seems there could be some kind of danger here.

Could it cause an electric shock somewhere (in general), or is it just dangerous in the sense that someone might think the circuit is off and go play with wires that might actually be live?

You are correct. The issues at hand are common maintenance shutoff (to solve the problem of which you speak) and common trip (to entirely remove power from a device with a problem).

• If the circuit serves a device that uses only 240V and does not supply neutral, then common maintenance shutoff will suffice. Go to an electrical supply house that deals in your panel/breaker type, and obtain a proper, listed handle-tie.
  

Proper, listed handle ties can be hard to find, especially on a Sunday. An adequate substitute for a handle-tie is a 2-pole breaker. It's overkill and costs about $8 more, but it works.

• If the circuit serves a 120/240V device (which takes 2 hots and neutral), then common trip is mandatory. Go to any hardware store and obtain a 2-pole breaker appropriate for your panel.

While you're at it, make sure they didn't slip you an alien breaker: a breaker not UL-listed or UL-classified for that panel. For instance if they slipped Homeline breakers into your Siemens panel, a Siemens handle-tie won't fit those breakers, now is the time to fix that.

Also, while you're labeling, put "GFCI protected" on any outlet fed off that GFCI. That's a code requirement most people overlook, causing home inspectors to "write them up".

You are correct that a 240 volt circuit with separate breakers on each leg is a hazard for future maintainers.

You should call the electrician and talk to him about it. You don't have to start by insisting that he come back, just ask him to tell you everything he remembers about the installation.

If he didn't have a reason for leaving the separate single pole breakers, or if you disagree with his reason, ask him to come back and finish the job properly.

If you can't reach an agreement with the electrician, you can probably install the double breaker yourself. This is something that a reasonably cautious homeowner can safely do.

At a minimum there should be a handle tie if it truly is a 240 V feed.

For a modern panel, in some cases, the handle ties cost 1/2 of what a brand new breaker costs in these cases. I prefer using a double pole breaker and now you have a couple of spare single poles. But yes, the handles should be tied with a listed tie or a double pole breaker is the code compliant method.

In addition to what others have mentioned, if there are two separate single-pole circuits, it's possible some future electrician might move one of the circuits, unaware they are both part of the same 240 V, and mistakenly shift the breaker down a spot. This will put both breakers on the same phase, and cause problems for your 240 V outlet.

However, are you sure your pool pump is actually 240 V? Is it possible the electrician was just running two completely unrelated circuits through the thicker cable? Maybe the pump is really just 120 V?