This is a circuit to split the output of a stereo headphone jack into seperate left and right line outputs. And to provide balanced audio outputs by using audio transformers.
If an unbalanced tip & sleeve jack is inserted, it bridges one side of the transformer to ground, providing an unbalanced feed without shorting anything out.
If only one output jack is connected, it'll get a mono sum of both feeds through the transformers being coupled together by the switches on the jacks. Both sockets are cross-coupled, so either can be used as the mono output. The arrows show the switch contacts that are made when the socket is empty, plugging a jack in disconnects them.
If a ground-lift feature is required, then instead of a common ground, all the grounds left of the transformers (the input side) should be connected together, and only to each other. And all the grounds on the right side of the transformers (the output side) should be connected together, and only to each other. With the ground-lift switch between the two halves. And either the input socket, or both output sockets, should be insulated from any metal enclosure that this device is fitted into. But a metal enclosure should be grounded with at least one of the sockets. And if your transformers have a shield, or a metal casing that can be grounded, they should be grounded to one side, too.
I tend to put the grounding on the mixer side of connections, making it the central grounding point for all my equipment. These days much equipment is double-insulated and unearthed (and theoretically shouldn't need a ground-lift switch). For instance, connecting a stage keyboard to a mixer, the keyboard may not be earthed and can't provide a ground to any shielding, it may actually induce a lot of noise on its shielding from a switch-mode power supply. But at the mixing side of things, you probably do have grounded equipment. And the mixer is always plugged in, external equipment might be unplugged at any time.
Think of it in the same way as dynamic microphones: They're an isolated sound source, with no earth. The only way to ground their leads is at the other end of their cable (at the console or patch bay).
Other people do enclosure grounding at the source side. Of course that depends on the source side actually being grounded. It often isn't.
Sometimes none of the equipment is earthed. Sometimes this isn't a problem. Sometimes it is, and equipment gets damaged when its being connected while something is powered (because floating power supplies can cause signal shielding to float at hundreds of volts). Some people deliberately add a mains earth ground somewhere to the chain, doing so at a patch panel is a reasonable approach. If doing so, it should be done at a permanent part of the installation (the patch panel, the mixer, the recorder, a monitor amplifier), so you can rely on knowing that it's always grounded. Experimentation may be needed to determine what point produces the quietest results.
The third option is to use plastic enclosures, use tightly twisted-pair wiring inside it, with shielding directly around all the wiring. This has an advantage of nothing inducing noise if it touches the enclosure, an often forgotten thing when people bung a DI box amongst equipment.
This is a traditional direct-inject (DI) circuit that doesn't require any active components, so doesn't need power. And can be fitted inside equipment or be an external interface. But because it uses transformers, shouldn't be placed next to magnetic noise sources (such as power transformers).