Can I power a two-phase home service with a single-phase generator?
Phases and poles

The question of "phases" comes up when dealing with 3-phase power.  The 3 phases are equally spaced: like runways at an airport arranged in a star or triangle pattern, one aiming north (12:00), southeast (4:00) and southwest (8:00).  

What if 2-phase power existed?  That would be runways aiming due north (12:00) and due south (6:00).  Wait a minute, that would be the same runway!   

So 2-phase power isn't really a thing; it's just single-phase power stacked on top of each other for at twice the voltage.  With a common middle (at neutral), it can be tapped end-middle for 120V or end-end for 240V.  And now you understand our 120V/240V split-phase system! 

Note that your generator has a NEMA 14 receptacle which is designed to power a split-phase system. 

You need a generator interlock.

Your setup must make it mechanically impossible to backfeed onto the electric grid.  Of course you'd have no reason to want to do that; so some people think "Oh, I can make it a checklist procedure", -- nope, not allowed.  Because that never works in a stressful emergency. 

Several companies make a variety of interlock kits with sliders or pivots that block the main breaker from being on at the same time as a double breaker in a particular position.  

You may notice that most regular breakers flip inward to be on.   The simplest interlock puts two breakers opposite each other, and inserts a spacer block so they can't both be inward (on). These two breakers are then back-fed, either from the utility or from the generator.  Siemens/Murray makes a $23 kit for their panels that also straps down those breakers, a Code requirement, and works with the covers removed, a Code requirement in some areas.  Square D also makes a very inexpensive one for their "QO" (not Homeline) panel line, but I'm not sure it provides all that.  However this only works on smaller panels where the utility side can be back-fed, e.g. Subpanels. 

Maybe put it in a subpanel

One easy way to provide the generator/interlock solution is to put a subpanel right next to your main panel, with one of the Siemens/Murray style interlocks on the top 2 breakers.  Then relocate each of your loads you want the generator to support into this panel.  There is no limit to the physical size of the subpanel; feel free to use a 42-space and put every circuit here.  This trick can even be used to solve a separate "my panel is full" problem!  That is why we are big fans of big panels.

All in all, such a subpanel is far, far cheaper than a "Reliance" style multi-circuit transfer switch, which allows you to switch up to 6 or 8 circuits between line and generator individually (why would you want that?).  It also place nicely with AFCI and GFCI.