CPI(M) said it wanted the probe to ascertain also if a "general atmosphere of intolerance being spread under patronage of BJP-led central government" has any role in the growth of an outfit, allegedly involved in Thursday's incident.
The Left party also alleged that the incident occurred due to a "very serious lapse" on the part of central and state intelligence agencies.
"The politburo of CPI(M) demands that a high powered judicial inquiry must be constituted immediately to probe into all these aspects, identify and punish the guilty.
The party said 22 of the total deceased belonged to the outfit, Swadhin Bharat Subash Sena (SBSS), which it added, had settled at Mathura Jawahar Bagh in 2014, allegedly piling up arms, ammunition and crude bombs preparing for a battle.
CPI(M) said it is "indeed inexplicable" that the SBSS engaged in these activities for two years at a place in close proximity to the office of district administration and barely two km away from Mathura Cantonment.