Q:

Does philosophy shed any light on how parties can fruitfully debate without an agreed source of truth?

A hallmark of recent political developments is extreme partisanship, where each side has near total distrust of the other. To exacerbate this situation there has been a breakdown in agreement over what constitutes the truth and even how one can reasonably arrive at that truth. Each side simply accuses the other of "fake news".
The side normally labelled as "left" or "liberal" generally evinces faith in something approximating rational methods of arriving at the truth. This approach dominated academia and discourse for much of the late 20th century. That which tends to be labelled "right" or "conservative" has now moved toward a more empirical understanding of truth. 
As a result, meaningful discourse between the two sides has become almost impossible, because each is using a different standard to support its arguments. Is there any philosophical thinking which might help resolve these disputes and allow the political discussion to move forward? Or is the gulf between the rational and empirical simply too great to bridge?

A:

From a purely epistemological point of view, no, philosophy is not really helpful. If anything, philosophy makes things worse. See this post and this post and the responses to it.
From a social point of view, Habermas' idea of "colonization of the life world" helps shed light on the problem and how to potentially solve it, but doesn't offer a full solution.    

For Habermas, communicative action is governed by practical rationality—ideas of social importance are mediated through the process of linguistic communication according to the rules of practical rationality. By contrast, technical rationality governs systems of instrumentality, like industries, or on a larger scale, the capitalist economy or the democratic political government. Ideas of instrumental importance to a system are mediated according to the rules of that system (the most obvious example is the capitalist economy's use of currency). Self-deception, and thus systematically distorted communication, is possible only when the lifeworld has been 'colonized' by instrumental rationality, so some social norm comes into existence and enjoys legitimate power even though it is not justifiable. This occurs when means of mediating instrumental ideas gains communicative power—as when someone pays a group of people to stay quiet during a public debate, or if financial or administrative resources are used to advertise some social viewpoint. When people take the resulting consensus as normatively relevant, the lifeworld has been colonized and communication has been systematically distorted. The 'colonization' metaphor is used because the use of steering media to arrive at social consensus is not native to the lifeworld—the decision-making processes of the systems world must encroach on the lifeworld in a way that is in a sense imperialistic:

In response to the comment by Gonzo:

you believe the gist of the extended quote is

I will try to illustrate by example:
One form of linguistic communication is practical rationality. If one were trying to advertise a product they are selling, this would be by listing the features of the product they are selling and explaining why it is better than competing products. 
Another form of communication is instrumental/technical: Here the person selling the product doesn't care about using reason or logic to communicate why their product is better, they simply want to sell as many products as possible (this is a technical or instrumental consideration). So they use the image of a scantly clad attractive person or maybe the face of a wall known personality to advertise for their product. 
The gist of the quote is: If the lifeworld (i.e. civil society, public discourse, etc...) is taken over by the second form of communication instead of the first one, then debate becomes irrational. 
This colonization occurs when individuals or organizations have the power or the means to use the second form of communication at scale. 

its salience in the context of the question,  

It should be obvious from the quote itself: 
"Self-deception, and thus systematically distorted communication, is possible only when the lifeworld has been 'colonized' by instrumental rationality, so some social norm comes into existence and enjoys legitimate power even though it is not justifiable."
-- for example the NRA has acquired means of instrumental communication, so that ideas like "We should arm teachers with hand guns" and "reducing access to guns won't reduce gun crimes" become legitimate debate points instead of irrational non-sense. 

[how] it sheds light on how to solve the posed problem.

This points to the reason that opposing parties can't even agree on what constitutes the truth. And helps shed light on "Is there any philosophical thinking which might help resolve these disputes and allow the political discussion to move forward?"  
It does not however address the question of "left = rationalism / right = empiricism" because the dichotomy posed in the OP is false. Both left and right have been guilty at times of ignoring either rational arguments or empirical evidence.

A:

Here are the questions regarding the extreme partisanship of politics today:

Is there any philosophical thinking which might help resolve these disputes and allow the political discussion to move forward? Or is the gulf between the rational and empirical simply too great to bridge?

One thing philosophy could do is clarify why the dispute is occurring. If that clarification is accurate it may provide suggestions for how the gulf might be bridged.
What might that clarification be? There are two possibilities, among perhaps many, that I will offer. The first is to consider viewing the world through the insights of moral foundations theory and the second considers viewing changes in the world through the insights of socionomics.
Graham, et. al., from the perspective of moral foundations theory, consider rationalism to be the position that thinks there is only one moral foundation. (page 4)

In this article we ask: How many “irreducible basic elements” are needed to represent, 
  understand, and explain the breadth of the moral domain? We use the term monist to describe scholars who assert that the answer is: one. This one is usually identified as justice or fairness, as Lawrence Kohlberg asserted: “Virtue is ultimately one, not many, and it is always the same ideal form regardless of climate or culture…The name of this ideal form is justice”

The others, the empiricists, such as Aristotle and William James hold a different opinion:

Other theorists—whom we’ll call pluralists—assert that the answer is: more than one.

Combine that with "individuals" supported mainly by the rationalist and there is a ground for tension where both sides in a political dispute can feel justified to rationalize that the other side is wrong and they are right.
That perspective suggests that perhaps engaging with people who disagree with us on a non-rational level that avoids the rationalizing may be one way to bridge the gulf.
However valuable that approach may be, it doesn't explain the dynamics of why the situation should be changing now.  Why are "recent political developments" so different from those in the past? What marks social periods when "each side has near total distrust of the other"?
That requires a theory of history. The problem with looking at history is we have a bias to thinking the trend should never change course. If the dinosaurs are ruling the earth they should always rule the earth, but that isn't how evolution portrays the past. It also isn't how markets behave. 
Socionomics looks for sociometers marking trends and hoping to find patterns that signal when trends will change. The most detailed sociometers are market price charts and socionomists have identified patterns of the wave behavior of price action to try to predict when a trend will change and what one can expect to see when the trend does change.
What makes this relevant is that a bullish trend has less extreme partisanship than a bearish trend. Couple that with a more or less evident wave structure and one might have an edge to predict trend changes. It sometimes works in trading the markets. Socionomists would claim it works more generally. 
Assuming either moral foundations theory or socionomics are correct, how would that understanding help us bridge the "gulf between the rational and empirical" sides in current social conflicts? 
In a bearish environment of negative social mood where righteous claims from both the rational and empirical groups become more intense, it may help one to recognize that perhaps the best one can do is to get out of the way and prepare for a change of trend in the opposite direction.

References
Graham, J., Haidt, J., Koleva, S., Motyl, M., Iyer, R., Wojcik, S. P., & Ditto, P. H. (2013). Moral foundations theory: The pragmatic validity of moral pluralism. In Advances in experimental social psychology (Vol. 47, pp. 55-130). Academic Press.
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2184440
Moral Foundations Theory, https://moralfoundations.org/
Socionomics Institute, https://www.socionomics.net/

