The letters concerned Senior Deputy Election Commissioner Maneesh Garg and an order he issued regarding the distribution of work among senior officials.
Sandhu and Joshi reportedly alleged that they and the officers concerned had not been consulted or informed before the changes were introduced.
Sandhu alleges unilateral allocation of work
In his letter, Sandhu alleged that Garg issued an order allocating work among Senior Deputy Election Commissioners, Deputy Election Commissioners, Director Generals and equivalent officers within hours of taking additional charge of the Administration Division on July 28.
Garg was assigned the additional charge while Deputy Election Commissioner Bhanu Prakash Yeuru was on leave.
According to NDTV, Sandhu wrote that the order was issued "without discussing or informing the revised allocation of work with the Commission or the undersigned".
Sandhu further alleged that Garg had assigned responsibilities to himself as well as to officers senior or equivalent to him.
"This has happened in spite of repeated and explicit oral and written instructions to all concerned officers by the undersigned against taking such actions without prior discussion/approval of the Commission," Sandhu wrote.
Joshi raises concerns over access to files
Apart from making similar allegations, Joshi reportedly stated in his letter that he had been denied access to relevant files when he sought them from the Administrative Division.
The letters reportedly clarified that they had been sent to the Cabinet Secretary in the commissioners' personal capacities and should not be treated as formal communications from the Commission as a whole.
ECI faces mounting scrutiny over electoral rolls
The latest development comes amid protests by Opposition parties, students and young people demanding the resignation of Chief Election Commissioner Gyanesh Kumar over alleged irregularities in electoral rolls.
The controversy follows a report by The Indian Express detailing differences within the three-member Election Commission involving Kumar and Election Commissioners Sandhu and Joshi.
According to the investigation, Sandhu and Joshi formally raised objections at least 14 times over a 10-month period regarding decisions and orders related to electoral rolls and the Special Intensive Revision (SIR) exercise.
They reportedly alleged that certain decisions had been taken without their knowledge.
The objections covered changes to Form 6, access to the electoral-roll database, the deletion and restoration of voter names, appeals concerning electoral rolls and aspects of the Commission's internal decision-making process.
The two commissioners also reportedly expressed concern over what they described as the gradual centralisation of the electoral-roll database.
The Election Commission has disputed the interpretation of the report, maintaining that written objections, suggestions and differing views are part of its internal deliberative process.