Do MOSFET relays for DC output always have a polarity specified?
DC-rated solid state relays are, well, for DC.  DC current is defined as the unidirectional flow of electric charge - in other words, current that only flows in one direction.  AC current is the bidirectional flow of charge - current that changes direction.  The distinction between the two is often misunderstood to be whether or not the current is wiggling or changing, but this is not how AC and DC current is defined.   You can have pulsed DC, and you can take an AC signal and bias it until it is DC (without making the signal any less 'wiggly').
So it isn't the DC solid state relays enforcing polarity.  You are enforcing that because you're specifying exactly that - a DC solid state relay.  When you specify a solid state relay that is DC, you are explicitly specifying that you want a solid state relay that has polarity and is only intended for applications where the direction of current does not change direction.  If your load requires reversing polarity (in other words, requires current flow in both directions/current that changes direction), then it is not a DC load.
You will find what you need if you look at solid state relays specified for AC and DC.  Be careful here, as there is another variety of solid-state relay that does not use MOSFETs, but rather uses SCRs/triacs (so-called 'zero-cross' AC solid state relays) and these will not work as expected for your load.  SSRs of this type will not turn off until the current reveres direction, so keep that gotcha in mind.
But any AC SSR using MOSFETs will work.  You should not need to change your current specifications as long as the relay specifies the current rating as RMS current.  This simply means it is rated for 'varying current that is equivalent to this many amps of constant/unchanging DC current'.  So 10A RMS just means it is rated for 10A of constant current, but also changing current as long as it averages to 10A as well.  This average is taken using the root mean square, so it accounts for losses scaling with \$ I^{2} \$.
If you want to be really really sure it will work for your load, just look at the construction of such SSRs:  They have two MOSFETs connected in series, source to source.  Thanks to the photodiode drive circuitry, the the drive voltage for the MOSFET gates is galvanically isolated.  The negative side of these diodes are connected to the junction between the two sources, while the positive side is connected to both gates.  This allows a single (isolated) voltage to turn on both MOSFETs, despite pointing opposite directions, simultaneously, as it is in series with both of the MOSFETs' sources.  Current of either polarity will flow through the matching MOSFET, and continue through the intrinsic body diode of the other MOSFET, and when the current switches directions, this process continues but the MOSFETs switch roles (as in which one's body diode is turned on).
Like this: 
It will work just fine for your load which requires both polarities.  Solid state relays of this construction have no contact polarity, and work even for things that seem like DC but sometimes need to switch directions, like your load.  They couldn't care less if your load is using 10A in one direction most of the time, or without changing.  This is also going to behave very similar to a real physical contact, albeit with more losses usually.
Post-answer note: often, there are those who have doubts that there is really a difference between true AC and pulsed DC.  If you are unconvinced, try powering the primary winding of most transformers with a sinusoidal AC waveform that has been biased so it is DC.  The waveform is identical, but it no longer crosses 0, swinging from 0 to 2*V instead of -V to V.  It will work as expected with AC current of course, but using the same waveform biased so it is DC will cause most transformers to rapidly saturate due to magnetic hysteresis in the core.  Likewise, forward converters require transformers to have a 'reset' winding which is wound in the opposite axial chirality (right handed vs left handed, like threads on a screw for example), which lets the DC driver circuit reverse the direction of the current by physically switching to a winding wound in the opposite direction instead of actually having to change the direction of current flow from the driver's point of view.
Even pulse transformers - which are specifically designed to work with DC only by using core materials with minimal hysteresis (they don't retain any magnetization after the current stops) - are rated by their volt-second integral as their critical specification.  If this is not carefully observed, or rather, balanced, the magnetic flux rapidly drops off due to saturation.  This ultimately dictates that the duty cycle must never exceed 50% so the transformer has time to 'reset' the magnetization of the core.
Even more simply, imagine you have a core-less electromagnet coil, a solenoid type winding.  If AC and pulsed DC are equivalent, then why does this coil magnetize your screwdriver if fed pulsed DC, yet demagnetize your screwdriver if fed AC pulsating at the same frequency as the pulsed DC?
Magnetism has a north and a south pole, and the symmetry is between them.
Switching between 'north and no field' is the same as switching between 'south and no field'.  But switching between either and no field is not the same as switching between north and south.
Just ask your screwdriver.