A regular household inverter cannot feed the grid. Period. Grid tie inverters have circuits and logic to syncronize the AC cycle to that of the grid, and adjust the voltage high enough to make the power go "into" the grid. Grid tie inverters are pure sine wave as well, to match the frequency and shape of the grid AC waveform.
Most household inverters are "modified sine wave" which means they are really a square wave with many steps. Not true AC voltage. Even if you happen to have a pure sine wave home inverter, its frequency stability could be off by a few percent, and the AC phase would never match that of the power grid.
Most grid tie solar systems (I have one) have some form of power company contract. Where one is paid for power produced, there is a separate meter showing what is fed to the grid, and you are paid based on that meter. Your house consumption is based on the other meter, billed at a different rate.
If you have a "net use" contract, that means you have one meter that runs forward or backwards, depending on house demand and what you generate. So if you generate more than you use, the meter gives you a credit. If you use more than you generate, the meter accumulates charges. Net effect for "net use" is to try and balance overall generation to overall use in kWH for a specific period of time (usually a year). If you generate more than you use, you do not get any carry forward or payment for a "net use" contract.
My contract is to get paid for generation, and the contract is for 20 years. The payment is way more than I'm charged for household use, in part to defray the capital costs of installing the system (government program). After 20 years, it becomes "net use".
As a note, most grid tie solar contracts prohibit the attachment of battery storage to their systems. That is to prevent people using the lower household rate to charge the batteries and feed that back to the grid at the higher payment rate. For "net use" systems, I would see no reason to ban batteries, as efficiency losses would prevent any benefit of cycling the same grid power to charge batteries...