I'm not HP savvy but I did some quick Googling (hp irf ring topology) and found a couple of docs that indicate it does support a ring topology.
Quote from page 5:
IRF topologies
Create an IRF virtual device in daisy chain topology,
or more reliably, ring topology, as shown in Figure 2. In ring chain
topology, the failure of one IRF link does not cause the IRF virtual
device to split as in daisy chain topology. Rather, the IRF virtual
device changes to a daisy chain topology without affecting network
services.
This page has the same quote so at least that seems like a possibility.
HP seems to be recommending the ring but beyond that I can't speak to HP gear.
I agree with @network_ninja but will extend it a bit.
How I'd solve this
Router1--L3--Router2
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Switch1--L2--Switch2
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PC1 PC2--------+
Router1 and Router2 are running VRRP, HSRP, GLBP or CARP to produce virtual default-GW IP address to the LAN.
This protocol will converse over the Switch core to agree which of the routers is owning the default-GW IP address at any given time.
PC2 is redundant linux server, which is using 'bonding' to redundantly connect to the Switches, it should be configured so that if the the virtual default-gw IP address stops responding to ARP WHO HAS, it'll switch to backup connection. IP address itself is not on the physical interfaces, but on the virtual bonding interface.
Equivalent solution is available to other OS, but often not included in base OS package.
PC1 is non-redundant server.
Switches are not running anything special, no spanning tree (as there is no L2 loop) and no LACP. They can be from different vendors and can be taken down for maintenance separately.
Routers are not running any switching, IP addresses are configured directly in the L3 interfaces facing the switches.
If you choose VRRP as your first-hop-redundancy-protocols, routers can be from different vendor. Each router can be taken down for maintenance separately, by gracefully switching VRRP priority before work on the primary.
Best Answer
End-user computers, generally, do not require this level of redundancy. You would see this level of fault tolerance in a stacked/chassis setup for something like a server farm. You would have two routing engines, two power supplies, multiple cards and multiple links to that node with 802.1ax (i.e. LACP/PAgP) to help protect you in every instance from an outage.
If you want redundancy, split the network into two. Have half your connections going to one switch, and the other half going to the former.
This is a very bad idea; not to mention it will not work how your anticipating it will. If you did split the pairs like this, you would only get use out of one set of them. Auto negotiation will allow a single pair to function by autoing down to 100Mb; remember, 100BASE-TX only functions on pairs 1,2,3 and 6.