It can be done, but not with cheap routers. And it can be tricky to set up.
The Internet itself is designed around “multi-homing”. most big networks have multiple connections to other networks (whether to ISPs or to other businesses that they choose to peer with). The BGP protocol is used worldwide to manage the dynamic network-network connections that comprise the Internet.
So one thing you could do (and I don’t recommend it, because it’s not easy to do if you’re not an IT/networking expert) is lease a static IP address block from an official registrar (not coming from one ISP’s address space) run BGP on your gateway router, advertising your network as being reachable via all of your upstream networks (various ISPs). You probably won’t be able to get IPv4 address space, but you should be able to get IPv6 space, since there is no shortage of IPv6 addresses.
But again, this kind of multi-homing is going to be overkill for most people, including small businesses.
Another option is to get a so-called “dual WAN” router, as @dsh1705 wrote. These let you connect to two upstream networks (e.g. cable/dsl/fiber modems from two service providers) and configure them as load balancing or active/standby operation.
In the load balancing mode, your outbound network connections will be directed to one or other other link. This will be on a per-connection (TCP session, mostly) basis. It will not attempt to distribute one session’s packets across multiple links because that would break the TCP protocol in many ways, creating outages.
In the active/standby mode, all your traffic will flow over the primary link until it fails, then all traffic will flow over the standby link. When the primary comes back up, the router might immediately switch back, or it might stay on the secondary until it fails (you should be able to select one or the other). One common use for this is to get a cellular modem to act as backup - you won’t want to use it when the primary network is operating, but this will keep you on-line (probably at reduced bandwidth and higher cost) when the primary fails.
But WRT the original question, using one for incoming and one for outgoing, that’s not a typical configuration.
If you have two active connections at once (e.g. the BGP method or a dual WAN router in the load-balance configuration), your network will receive packets sent to your network on either interface (via your own addresses if you use the BGP method or via your ISPs’ assigned addresses if you use the dual WAN router), assuming your firewall configuration doesn’t block them. And you can (or at least should be able to) configure priorities to favor one link over the other for your own outbound connections.
But this is going to be on a per-connection (e.g. TCP session) basis. You absolutely can not do it on a per-packet basis (e.g. receive everything on one link, send everything on another) because doing that will break the TCP protocol, resulting in either dropped packets (lost connections) or out-of-order packet delivery, resulting in massive slowdowns (because TCP will assume this is due to network congestion and slow itself in response).
Consumer satellite Internet access sometimes offer this behavior (outbound packets going over a dial-up or DSL line, and inbound packets arriving over the satellite), but that’s a special case because a single network (run by the satellite provider) is managing the other end of both connections, so everything can be synchronized properly in that situation.
I’m not sure how much of this will help, but I hope it does.